| L(s) = 1 | − 1.74e3·3-s − 1.56e5·5-s + 2.39e6·7-s − 6.89e6·9-s − 7.76e6·11-s − 7.86e7·13-s + 2.71e8·15-s + 1.00e9·17-s − 1.25e9·19-s − 4.16e9·21-s − 2.85e10·23-s + 1.83e10·25-s + 4.28e9·27-s − 8.18e9·29-s − 2.84e11·31-s + 1.35e10·33-s − 3.73e11·35-s − 1.27e12·37-s + 1.36e11·39-s − 1.93e12·41-s − 1.49e12·43-s + 1.07e12·45-s + 7.27e11·47-s − 4.92e12·49-s − 1.74e12·51-s + 5.40e12·53-s + 1.21e12·55-s + ⋯ |
| L(s) = 1 | − 0.459·3-s − 0.894·5-s + 1.09·7-s − 0.480·9-s − 0.120·11-s − 0.347·13-s + 0.410·15-s + 0.594·17-s − 0.322·19-s − 0.504·21-s − 1.74·23-s + 3/5·25-s + 0.0788·27-s − 0.0881·29-s − 1.85·31-s + 0.0552·33-s − 0.982·35-s − 2.21·37-s + 0.159·39-s − 1.55·41-s − 0.838·43-s + 0.429·45-s + 0.209·47-s − 1.03·49-s − 0.272·51-s + 0.631·53-s + 0.107·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{7} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 580 p T + 122458 p^{4} T^{2} + 580 p^{16} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 341860 p T + 217349292690 p^{2} T^{2} - 341860 p^{16} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7767360 T + 523131456990482 p T^{2} + 7767360 p^{15} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 78626540 T + 5358436553761446 p T^{2} + 78626540 p^{15} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1005018420 T + 5977228326554728582 T^{2} - 1005018420 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1254824792 T + 27243202793036351814 T^{2} + 1254824792 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 28543436940 T + \)\(69\!\cdots\!10\)\( T^{2} + 28543436940 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 282405588 p T + \)\(14\!\cdots\!74\)\( T^{2} + 282405588 p^{16} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 284875900088 T + \)\(64\!\cdots\!38\)\( T^{2} + 284875900088 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1279483375820 T + \)\(99\!\cdots\!10\)\( T^{2} + 1279483375820 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1935823797588 T + \)\(20\!\cdots\!38\)\( T^{2} + 1935823797588 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1494680579420 T + \)\(59\!\cdots\!14\)\( T^{2} + 1494680579420 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 727376141100 T - \)\(57\!\cdots\!98\)\( T^{2} - 727376141100 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5401096800420 T + \)\(14\!\cdots\!90\)\( T^{2} - 5401096800420 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 46263127281096 T + \)\(12\!\cdots\!02\)\( T^{2} - 46263127281096 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4426392354004 T - \)\(76\!\cdots\!94\)\( T^{2} - 4426392354004 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 49372806203980 T + \)\(17\!\cdots\!50\)\( T^{2} - 49372806203980 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 61416965094936 T + \)\(85\!\cdots\!26\)\( T^{2} - 61416965094936 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 73991889212260 T + \)\(18\!\cdots\!18\)\( T^{2} - 73991889212260 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 265264590947216 T + \)\(61\!\cdots\!62\)\( T^{2} + 265264590947216 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 192322586199900 T + \)\(89\!\cdots\!70\)\( T^{2} + 192322586199900 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 597697533525228 T + \)\(43\!\cdots\!94\)\( T^{2} + 597697533525228 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1078510743819260 T + \)\(12\!\cdots\!50\)\( T^{2} + 1078510743819260 p^{15} T^{3} + p^{30} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.27732681158287957408209169284, −14.12185716928036396545012630557, −12.90394244835362697306499896295, −12.28656655529501856505303422845, −11.49998409041688055410293724875, −11.44776677899084701802565789112, −10.44888443503401604149492570908, −9.837213844429777609218389725879, −8.433764801624987371520013356235, −8.392066223604943272915074610261, −7.41628051131474322657524222305, −6.73428630758542265229391305578, −5.43218243895507711847312815459, −5.21584329495773985465549122784, −4.04657802441929715114873301115, −3.46223225836833578646504907180, −2.14160324079772523999184517465, −1.44802771167174606501387415612, 0, 0,
1.44802771167174606501387415612, 2.14160324079772523999184517465, 3.46223225836833578646504907180, 4.04657802441929715114873301115, 5.21584329495773985465549122784, 5.43218243895507711847312815459, 6.73428630758542265229391305578, 7.41628051131474322657524222305, 8.392066223604943272915074610261, 8.433764801624987371520013356235, 9.837213844429777609218389725879, 10.44888443503401604149492570908, 11.44776677899084701802565789112, 11.49998409041688055410293724875, 12.28656655529501856505303422845, 12.90394244835362697306499896295, 14.12185716928036396545012630557, 14.27732681158287957408209169284
