| L(s) = 1 | + 2.43·2-s + 3.91·4-s − 1.77·5-s + 1.33·7-s + 4.64·8-s − 4.31·10-s + 2.57·11-s − 6.99·13-s + 3.23·14-s + 3.47·16-s − 5.20·19-s − 6.93·20-s + 6.27·22-s + 0.415·23-s − 1.85·25-s − 17.0·26-s + 5.20·28-s − 7.64·29-s + 7.09·31-s − 0.850·32-s − 2.36·35-s − 0.225·37-s − 12.6·38-s − 8.24·40-s − 1.46·41-s − 6.92·43-s + 10.0·44-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.95·4-s − 0.793·5-s + 0.503·7-s + 1.64·8-s − 1.36·10-s + 0.777·11-s − 1.94·13-s + 0.864·14-s + 0.867·16-s − 1.19·19-s − 1.55·20-s + 1.33·22-s + 0.0865·23-s − 0.370·25-s − 3.33·26-s + 0.983·28-s − 1.41·29-s + 1.27·31-s − 0.150·32-s − 0.399·35-s − 0.0371·37-s − 2.05·38-s − 1.30·40-s − 0.228·41-s − 1.05·43-s + 1.52·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 + 6.99T + 13T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 - 0.415T + 23T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 + 0.225T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 - 8.21T + 47T^{2} \) |
| 53 | \( 1 + 9.57T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 2.51T + 67T^{2} \) |
| 71 | \( 1 + 8.93T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 7.84T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 0.540T + 97T^{2} \) |
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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
−7.32822940437040994162740606175, −6.68850276188833152537528001059, −6.00914981203965493301985357380, −5.13260052899305119160212515830, −4.59311985858395419775677940817, −4.12583196050816914670633889068, −3.36473812032616696342967461932, −2.47872519666257605611584913029, −1.76559193556319819950263203137, 0,
1.76559193556319819950263203137, 2.47872519666257605611584913029, 3.36473812032616696342967461932, 4.12583196050816914670633889068, 4.59311985858395419775677940817, 5.13260052899305119160212515830, 6.00914981203965493301985357380, 6.68850276188833152537528001059, 7.32822940437040994162740606175
