| L(s) = 1 | + 0.216·2-s + 2.05·3-s − 1.95·4-s − 4.07·5-s + 0.444·6-s − 1.74·7-s − 0.855·8-s + 1.22·9-s − 0.881·10-s − 4.97·11-s − 4.01·12-s + 4.99·13-s − 0.376·14-s − 8.37·15-s + 3.72·16-s + 1.60·17-s + 0.265·18-s − 8.02·19-s + 7.95·20-s − 3.58·21-s − 1.07·22-s − 5.45·23-s − 1.75·24-s + 11.5·25-s + 1.08·26-s − 3.64·27-s + 3.40·28-s + ⋯ |
| L(s) = 1 | + 0.153·2-s + 1.18·3-s − 0.976·4-s − 1.82·5-s + 0.181·6-s − 0.658·7-s − 0.302·8-s + 0.409·9-s − 0.278·10-s − 1.49·11-s − 1.15·12-s + 1.38·13-s − 0.100·14-s − 2.16·15-s + 0.930·16-s + 0.388·17-s + 0.0626·18-s − 1.84·19-s + 1.77·20-s − 0.781·21-s − 0.229·22-s − 1.13·23-s − 0.359·24-s + 2.31·25-s + 0.212·26-s − 0.701·27-s + 0.642·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 349 | \( 1 + T \) |
| good | 2 | \( 1 - 0.216T + 2T^{2} \) |
| 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 4.07T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 + 8.02T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 - 0.688T + 29T^{2} \) |
| 31 | \( 1 - 9.28T + 31T^{2} \) |
| 37 | \( 1 + 2.03T + 37T^{2} \) |
| 41 | \( 1 - 4.03T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 + 6.84T + 47T^{2} \) |
| 53 | \( 1 - 5.31T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 + 3.62T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 1.62T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 3.93T + 89T^{2} \) |
| 97 | \( 1 + 8.97T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.95205189272260844591995652583, −10.04782615781932691232600553711, −8.736353667670824571014914249034, −8.245992621055060884728880930779, −7.81502974673948475288458774972, −6.19082672328400088699836234590, −4.54915401541692942061563296124, −3.75002897582300352504534376014, −2.96344889369968057674732553302, 0,
2.96344889369968057674732553302, 3.75002897582300352504534376014, 4.54915401541692942061563296124, 6.19082672328400088699836234590, 7.81502974673948475288458774972, 8.245992621055060884728880930779, 8.736353667670824571014914249034, 10.04782615781932691232600553711, 10.95205189272260844591995652583
