| L(s) = 1 | − 4·2-s + 12·4-s − 9·5-s + 19·7-s − 32·8-s + 36·10-s − 24·11-s + 61·13-s − 76·14-s + 80·16-s + 3·17-s + 133·19-s − 108·20-s + 96·22-s + 69·23-s + 47·25-s − 244·26-s + 228·28-s + 237·29-s + 211·31-s − 192·32-s − 12·34-s − 171·35-s + 262·37-s − 532·38-s + 288·40-s + 468·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.804·5-s + 1.02·7-s − 1.41·8-s + 1.13·10-s − 0.657·11-s + 1.30·13-s − 1.45·14-s + 5/4·16-s + 0.0428·17-s + 1.60·19-s − 1.20·20-s + 0.930·22-s + 0.625·23-s + 0.375·25-s − 1.84·26-s + 1.53·28-s + 1.51·29-s + 1.22·31-s − 1.06·32-s − 0.0605·34-s − 0.825·35-s + 1.16·37-s − 2.27·38-s + 1.13·40-s + 1.78·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.324731238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.324731238\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 9 T + 34 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 19 T + 540 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 24 T + 1861 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 61 T + 5088 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 9592 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 p T + 12234 T^{2} - 7 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 p T + 25288 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 237 T + 51244 T^{2} - 237 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 211 T + 68586 T^{2} - 211 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 262 T + 72162 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 468 T + 191653 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 p T + 24783 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 483 T + 212812 T^{2} - 483 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 150 T + 257074 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 168 T + 416869 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1049 T + 675906 T^{2} + 1049 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1166 T + 907395 T^{2} + 1166 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 312 T + 498238 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 311 T + 698028 T^{2} + 311 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 349 T + 976602 T^{2} - 349 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1221 T + 1412098 T^{2} - 1221 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 492 T + 1092454 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 128 T + 1715097 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.21695908929644251083219027142, −12.03571401144105409218043838421, −11.42916099519757742877233420313, −11.14991693174601643398692290076, −10.51743045390285681073524494457, −10.36453009518044151957206098776, −9.383491822973620437707586871926, −9.124138357204641761195068770497, −8.254859267019606581004661126327, −8.250448239662948276952832606410, −7.43405464737809151935461209247, −7.39637503049579507498152148834, −6.27923160804053056309974071552, −5.89120147038708731964916913319, −4.91194099336879025209143242706, −4.32853905527482945936387563174, −3.21010623684492776275209463611, −2.65076329714025207265160829638, −1.28632287029265319617425772286, −0.812327752269083132670016101859,
0.812327752269083132670016101859, 1.28632287029265319617425772286, 2.65076329714025207265160829638, 3.21010623684492776275209463611, 4.32853905527482945936387563174, 4.91194099336879025209143242706, 5.89120147038708731964916913319, 6.27923160804053056309974071552, 7.39637503049579507498152148834, 7.43405464737809151935461209247, 8.250448239662948276952832606410, 8.254859267019606581004661126327, 9.124138357204641761195068770497, 9.383491822973620437707586871926, 10.36453009518044151957206098776, 10.51743045390285681073524494457, 11.14991693174601643398692290076, 11.42916099519757742877233420313, 12.03571401144105409218043838421, 12.21695908929644251083219027142
