| L(s) = 1 | + 0.938·2-s − 1.11·4-s − 5-s − 2.34·7-s − 2.92·8-s − 0.938·10-s + 1.48·11-s − 13-s − 2.20·14-s − 0.512·16-s − 2.30·17-s + 8.20·19-s + 1.11·20-s + 1.39·22-s + 7.19·23-s + 25-s − 0.938·26-s + 2.62·28-s − 4.07·29-s + 9.14·31-s + 5.37·32-s − 2.16·34-s + 2.34·35-s − 7.18·37-s + 7.70·38-s + 2.92·40-s + 7.75·41-s + ⋯ |
| L(s) = 1 | + 0.663·2-s − 0.559·4-s − 0.447·5-s − 0.885·7-s − 1.03·8-s − 0.296·10-s + 0.448·11-s − 0.277·13-s − 0.588·14-s − 0.128·16-s − 0.559·17-s + 1.88·19-s + 0.250·20-s + 0.297·22-s + 1.50·23-s + 0.200·25-s − 0.184·26-s + 0.495·28-s − 0.757·29-s + 1.64·31-s + 0.950·32-s − 0.371·34-s + 0.396·35-s − 1.18·37-s + 1.25·38-s + 0.462·40-s + 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.938T + 2T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 8.20T + 19T^{2} \) |
| 23 | \( 1 - 7.19T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 9.14T + 31T^{2} \) |
| 37 | \( 1 + 7.18T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 + 0.701T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 5.59T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 - 1.91T + 97T^{2} \) |
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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
−7.77668888715662909670834941789, −6.96654586226590012509051835732, −6.40298549151647632649574668527, −5.49036313794505813337277710945, −4.87430945949884596142142040073, −4.15506080310393054552504778551, −3.19305063241617695409228851902, −2.97414258628594386889601539172, −1.20479534435658248697126029113, 0,
1.20479534435658248697126029113, 2.97414258628594386889601539172, 3.19305063241617695409228851902, 4.15506080310393054552504778551, 4.87430945949884596142142040073, 5.49036313794505813337277710945, 6.40298549151647632649574668527, 6.96654586226590012509051835732, 7.77668888715662909670834941789
