| L(s) = 1 | + 56·3-s + 14·5-s + 1.84e3·7-s + 4.20e3·9-s + 2.40e3·11-s + 2.14e4·13-s + 784·15-s + 3.50e4·17-s − 2.40e3·19-s + 1.03e5·21-s − 6.16e4·23-s − 2.97e4·25-s + 2.86e5·27-s − 1.91e5·29-s − 1.66e5·31-s + 1.34e5·33-s + 2.58e4·35-s + 5.59e5·37-s + 1.20e6·39-s + 1.61e6·41-s − 5.35e5·43-s + 5.89e4·45-s + 1.76e6·47-s + 1.65e6·49-s + 1.96e6·51-s − 2.31e6·53-s + 3.37e4·55-s + ⋯ |
| L(s) = 1 | + 1.19·3-s + 0.0500·5-s + 2.03·7-s + 1.92·9-s + 0.545·11-s + 2.70·13-s + 0.0599·15-s + 1.73·17-s − 0.0805·19-s + 2.43·21-s − 1.05·23-s − 0.380·25-s + 2.80·27-s − 1.45·29-s − 1.00·31-s + 0.653·33-s + 0.101·35-s + 1.81·37-s + 3.24·39-s + 3.65·41-s − 1.02·43-s + 0.0963·45-s + 2.48·47-s + 2.01·49-s + 2.07·51-s − 2.13·53-s + 0.0273·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(34.25918190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(34.25918190\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 264 p T + 5122 p^{3} T^{2} - 264 p^{8} T^{3} + p^{14} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 56 T - 1073 T^{2} + 3080 p T^{3} + 674920 p^{2} T^{4} + 3080 p^{8} T^{5} - 1073 p^{14} T^{6} - 56 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 14 T + 29901 T^{2} + 520674 p T^{3} - 209457644 p^{2} T^{4} + 520674 p^{8} T^{5} + 29901 p^{14} T^{6} - 14 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2408 T + 10061967 T^{2} + 104116730760 T^{3} - 474568035257576 T^{4} + 104116730760 p^{7} T^{5} + 10061967 p^{14} T^{6} - 2408 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 10724 T + 152410814 T^{2} - 10724 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 35098 T + 212929257 T^{2} - 6958634809098 T^{3} + 364377865570683988 T^{4} - 6958634809098 p^{7} T^{5} + 212929257 p^{14} T^{6} - 35098 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2408 T - 1156135049 T^{2} - 1506950395720 T^{3} + 545899719750428152 T^{4} - 1506950395720 p^{7} T^{5} - 1156135049 p^{14} T^{6} + 2408 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 61684 T - 2114477901 T^{2} - 54914621238708 T^{3} + 10491311795669237608 T^{4} - 54914621238708 p^{7} T^{5} - 2114477901 p^{14} T^{6} + 61684 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 95660 T + 34197541822 T^{2} + 95660 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 166012 T - 1067536483 p T^{2} + 934379765212740 T^{3} + \)\(21\!\cdots\!84\)\( T^{4} + 934379765212740 p^{7} T^{5} - 1067536483 p^{15} T^{6} + 166012 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 559814 T + 77524081237 T^{2} - 25753615570568702 T^{3} + \)\(16\!\cdots\!24\)\( T^{4} - 25753615570568702 p^{7} T^{5} + 77524081237 p^{14} T^{6} - 559814 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 805980 T + 544490570278 T^{2} - 805980 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 6232 p T + 534162600534 T^{2} + 6232 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1769292 T + 1340647882547 T^{2} - 1373855340249153972 T^{3} + \)\(13\!\cdots\!88\)\( T^{4} - 1373855340249153972 p^{7} T^{5} + 1340647882547 p^{14} T^{6} - 1769292 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 2317194 T + 2139024918053 T^{2} + 2041310819323319346 T^{3} + \)\(27\!\cdots\!88\)\( T^{4} + 2041310819323319346 p^{7} T^{5} + 2139024918053 p^{14} T^{6} + 2317194 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 660352 T - 4531445555481 T^{2} + 6466596179869056 T^{3} + \)\(17\!\cdots\!88\)\( T^{4} + 6466596179869056 p^{7} T^{5} - 4531445555481 p^{14} T^{6} - 660352 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1463042 T - 2995701633619 T^{2} - 1681462677906192678 T^{3} + \)\(97\!\cdots\!04\)\( T^{4} - 1681462677906192678 p^{7} T^{5} - 2995701633619 p^{14} T^{6} + 1463042 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1784280 T - 5420427629185 T^{2} - 6275920241430481080 T^{3} + \)\(18\!\cdots\!96\)\( T^{4} - 6275920241430481080 p^{7} T^{5} - 5420427629185 p^{14} T^{6} + 1784280 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 274400 T - 1862728669394 T^{2} + 274400 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4549062 T - 5976062559175 T^{2} + 20813007667450176150 T^{3} + \)\(36\!\cdots\!84\)\( T^{4} + 20813007667450176150 p^{7} T^{5} - 5976062559175 p^{14} T^{6} + 4549062 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8673964 T + 24597682454603 T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!04\)\( T^{4} + \)\(10\!\cdots\!00\)\( p^{7} T^{5} + 24597682454603 p^{14} T^{6} + 8673964 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 10594360 T + 68478299343430 T^{2} - 10594360 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 1779750 T - 23603446896199 T^{2} - \)\(10\!\cdots\!50\)\( T^{3} - \)\(13\!\cdots\!40\)\( T^{4} - \)\(10\!\cdots\!50\)\( p^{7} T^{5} - 23603446896199 p^{14} T^{6} + 1779750 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 1748964 T + 72590049947446 T^{2} + 1748964 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−8.670071585118868809731097290439, −8.401772269957105519986694163516, −7.85489616837917819192432296874, −7.84426865139574111909616550133, −7.51289288117623184292945541352, −7.37621860062251855269795182283, −7.19078685226657268864656690411, −6.28028768096994226734287621772, −6.13235558697110103356457587253, −5.96236838076228912926223502571, −5.73515543423908211931083506663, −5.26416026578023570100250404670, −4.52977184293590814378155981093, −4.51345708171829566470826947184, −4.28500277298336204006802927049, −3.79938210199151210583213155027, −3.52229771399934831796976343184, −3.32906670482530602455260132359, −2.69816561637934675650291563587, −2.19472175431953741923489999809, −1.73064351829055000806950376415, −1.66598793371003698054693666892, −1.21707929115132173591256957364, −0.806425467189971276717523388570, −0.73454415030337869311993039489,
0.73454415030337869311993039489, 0.806425467189971276717523388570, 1.21707929115132173591256957364, 1.66598793371003698054693666892, 1.73064351829055000806950376415, 2.19472175431953741923489999809, 2.69816561637934675650291563587, 3.32906670482530602455260132359, 3.52229771399934831796976343184, 3.79938210199151210583213155027, 4.28500277298336204006802927049, 4.51345708171829566470826947184, 4.52977184293590814378155981093, 5.26416026578023570100250404670, 5.73515543423908211931083506663, 5.96236838076228912926223502571, 6.13235558697110103356457587253, 6.28028768096994226734287621772, 7.19078685226657268864656690411, 7.37621860062251855269795182283, 7.51289288117623184292945541352, 7.84426865139574111909616550133, 7.85489616837917819192432296874, 8.401772269957105519986694163516, 8.670071585118868809731097290439
