| L(s) = 1 | − 16·2-s + 56·3-s + 64·4-s + 238·5-s − 896·6-s + 168·7-s + 1.02e3·8-s + 2.76e3·9-s − 3.80e3·10-s − 5.84e3·11-s + 3.58e3·12-s + 2.63e3·13-s − 2.68e3·14-s + 1.33e4·15-s − 1.63e4·16-s + 4.76e4·17-s − 4.43e4·18-s + 4.10e4·19-s + 1.52e4·20-s + 9.40e3·21-s + 9.35e4·22-s + 4.93e4·23-s + 5.73e4·24-s + 1.60e5·25-s − 4.21e4·26-s + 4.46e4·27-s + 1.07e4·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.19·3-s + 1/2·4-s + 0.851·5-s − 1.69·6-s + 0.185·7-s + 0.707·8-s + 1.26·9-s − 1.20·10-s − 1.32·11-s + 0.598·12-s + 0.332·13-s − 0.261·14-s + 1.01·15-s − 16-s + 2.35·17-s − 1.79·18-s + 1.37·19-s + 0.425·20-s + 0.221·21-s + 1.87·22-s + 0.845·23-s + 0.846·24-s + 2.05·25-s − 0.469·26-s + 0.436·27-s + 0.0925·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.952386763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.952386763\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 24 p T - 638 p^{3} T^{2} - 24 p^{8} T^{3} + p^{14} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 56 T + 367 T^{2} + 29960 p T^{3} - 441560 p^{2} T^{4} + 29960 p^{8} T^{5} + 367 p^{14} T^{6} - 56 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 238 T - 104211 T^{2} - 219198 p T^{3} + 613820116 p^{2} T^{4} - 219198 p^{8} T^{5} - 104211 p^{14} T^{6} - 238 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 5848 T - 1200723 p T^{2} + 49314517320 T^{3} + 1201235432123224 T^{4} + 49314517320 p^{7} T^{5} - 1200723 p^{15} T^{6} + 5848 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 1316 T + 34154174 T^{2} - 1316 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 47642 T + 1043139177 T^{2} - 19339966944522 T^{3} + 404850845465430868 T^{4} - 19339966944522 p^{7} T^{5} + 1043139177 p^{14} T^{6} - 47642 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 41048 T - 382971689 T^{2} - 11500275107720 T^{3} + 1971126321462185272 T^{4} - 11500275107720 p^{7} T^{5} - 382971689 p^{14} T^{6} - 41048 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 49316 T - 3271709901 T^{2} + 54537239624292 T^{3} + 12659477851769669608 T^{4} + 54537239624292 p^{7} T^{5} - 3271709901 p^{14} T^{6} - 49316 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 172820 T + 40166510782 T^{2} - 172820 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 70252 T - 22773142333 T^{2} + 1919055786031020 T^{3} - \)\(12\!\cdots\!96\)\( T^{4} + 1919055786031020 p^{7} T^{5} - 22773142333 p^{14} T^{6} - 70252 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 88166 T + 103041424117 T^{2} + 25138942165957282 T^{3} - \)\(67\!\cdots\!16\)\( T^{4} + 25138942165957282 p^{7} T^{5} + 103041424117 p^{14} T^{6} - 88166 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 1156260 T + 721703173798 T^{2} + 1156260 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 57544 T + 389112928854 T^{2} - 57544 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1412292 T + 731963410547 T^{2} - 352167779849094972 T^{3} + \)\(32\!\cdots\!88\)\( T^{4} - 352167779849094972 p^{7} T^{5} + 731963410547 p^{14} T^{6} - 1412292 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2361174 T + 1840468460933 T^{2} - 3270820811450183406 T^{3} + \)\(60\!\cdots\!68\)\( T^{4} - 3270820811450183406 p^{7} T^{5} + 1840468460933 p^{14} T^{6} - 2361174 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1842512 T - 1854167432121 T^{2} + 500638023944439024 T^{3} + \)\(12\!\cdots\!28\)\( T^{4} + 500638023944439024 p^{7} T^{5} - 1854167432121 p^{14} T^{6} + 1842512 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1278242 T - 2859497061619 T^{2} - 2290719592631767878 T^{3} + \)\(51\!\cdots\!04\)\( T^{4} - 2290719592631767878 p^{7} T^{5} - 2859497061619 p^{14} T^{6} + 1278242 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2121480 T - 3508287504865 T^{2} + 8724498066914483880 T^{3} + \)\(13\!\cdots\!96\)\( T^{4} + 8724498066914483880 p^{7} T^{5} - 3508287504865 p^{14} T^{6} - 2121480 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3600160 T + 20950979040046 T^{2} - 3600160 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1634682 T - 14057803596295 T^{2} + 8769755399259278550 T^{3} + \)\(12\!\cdots\!24\)\( T^{4} + 8769755399259278550 p^{7} T^{5} - 14057803596295 p^{14} T^{6} - 1634682 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 9192604 T + 25128491333003 T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!44\)\( T^{4} - \)\(19\!\cdots\!80\)\( p^{7} T^{5} + 25128491333003 p^{14} T^{6} - 9192604 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 28280 T + 42919362781510 T^{2} + 28280 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8936550 T - 24286759552519 T^{2} + \)\(14\!\cdots\!50\)\( T^{3} + \)\(53\!\cdots\!20\)\( T^{4} + \)\(14\!\cdots\!50\)\( p^{7} T^{5} - 24286759552519 p^{14} T^{6} + 8936550 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 5874204 T + 167733185963446 T^{2} - 5874204 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.37593109711960404682013509539, −13.22334667468482983806600491608, −12.24784600466749305234349563932, −12.20394890856757011670640074478, −12.02578565215134253898458953301, −10.80553633908736571166543719922, −10.74964288509576927491975356349, −10.26846222914876209033288555047, −9.964668166660463024184031814782, −9.661066970369448145780138292809, −9.378827565290466004694295664174, −8.558999980883254884409565695148, −8.401395713794694159538445818033, −8.220474736804353486517114638871, −7.61522673375439852297053696182, −6.93589192782959242236176041601, −6.90797301692274462850428231176, −5.69933867205588996743180483269, −4.99384398925136809305952841458, −4.98274849897221237012060659966, −3.49235206994215538916097971359, −3.08486100807694147698516493363, −2.29469491037461225374195697173, −1.10464563684626396521844490985, −1.00219895995496005006123827351,
1.00219895995496005006123827351, 1.10464563684626396521844490985, 2.29469491037461225374195697173, 3.08486100807694147698516493363, 3.49235206994215538916097971359, 4.98274849897221237012060659966, 4.99384398925136809305952841458, 5.69933867205588996743180483269, 6.90797301692274462850428231176, 6.93589192782959242236176041601, 7.61522673375439852297053696182, 8.220474736804353486517114638871, 8.401395713794694159538445818033, 8.558999980883254884409565695148, 9.378827565290466004694295664174, 9.661066970369448145780138292809, 9.964668166660463024184031814782, 10.26846222914876209033288555047, 10.74964288509576927491975356349, 10.80553633908736571166543719922, 12.02578565215134253898458953301, 12.20394890856757011670640074478, 12.24784600466749305234349563932, 13.22334667468482983806600491608, 13.37593109711960404682013509539
