| L(s) = 1 | + 2-s + 1.44·3-s + 4-s − 0.733·5-s + 1.44·6-s − 7-s + 8-s − 0.912·9-s − 0.733·10-s − 0.799·11-s + 1.44·12-s − 4.74·13-s − 14-s − 1.05·15-s + 16-s + 6.38·17-s − 0.912·18-s − 0.733·20-s − 1.44·21-s − 0.799·22-s − 8.18·23-s + 1.44·24-s − 4.46·25-s − 4.74·26-s − 5.65·27-s − 28-s + 6.50·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.834·3-s + 0.5·4-s − 0.327·5-s + 0.589·6-s − 0.377·7-s + 0.353·8-s − 0.304·9-s − 0.231·10-s − 0.240·11-s + 0.417·12-s − 1.31·13-s − 0.267·14-s − 0.273·15-s + 0.250·16-s + 1.54·17-s − 0.215·18-s − 0.163·20-s − 0.315·21-s − 0.170·22-s − 1.70·23-s + 0.294·24-s − 0.892·25-s − 0.930·26-s − 1.08·27-s − 0.188·28-s + 1.20·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 + 0.733T + 5T^{2} \) |
| 11 | \( 1 + 0.799T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 - 6.38T + 17T^{2} \) |
| 23 | \( 1 + 8.18T + 23T^{2} \) |
| 29 | \( 1 - 6.50T + 29T^{2} \) |
| 31 | \( 1 + 2.93T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 8.05T + 41T^{2} \) |
| 43 | \( 1 - 1.98T + 43T^{2} \) |
| 47 | \( 1 + 0.410T + 47T^{2} \) |
| 53 | \( 1 + 6.93T + 53T^{2} \) |
| 59 | \( 1 + 6.25T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 9.93T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 3.22T + 73T^{2} \) |
| 79 | \( 1 + 5.56T + 79T^{2} \) |
| 83 | \( 1 - 0.511T + 83T^{2} \) |
| 89 | \( 1 + 5.55T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.65364051674133336903005359905, −7.52571604580236209460018497200, −6.20350551064209639024041863958, −5.77714947958574783736892710607, −4.79995865924706183811446833064, −4.05990881811628495078579179554, −3.18056380266124993101865190320, −2.73180515775055965351274684465, −1.73784506221736067375986294003, 0,
1.73784506221736067375986294003, 2.73180515775055965351274684465, 3.18056380266124993101865190320, 4.05990881811628495078579179554, 4.79995865924706183811446833064, 5.77714947958574783736892710607, 6.20350551064209639024041863958, 7.52571604580236209460018497200, 7.65364051674133336903005359905
