| L(s) = 1 | − 2-s + 4-s − 0.585·5-s + 4.82·7-s − 8-s + 0.585·10-s − 4·11-s + 2.82·13-s − 4.82·14-s + 16-s − 6.82·19-s − 0.585·20-s + 4·22-s − 3.17·23-s − 4.65·25-s − 2.82·26-s + 4.82·28-s + 0.585·29-s − 0.828·31-s − 32-s − 2.82·35-s + 2.24·37-s + 6.82·38-s + 0.585·40-s + 3.07·41-s + 1.65·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.261·5-s + 1.82·7-s − 0.353·8-s + 0.185·10-s − 1.20·11-s + 0.784·13-s − 1.29·14-s + 0.250·16-s − 1.56·19-s − 0.130·20-s + 0.852·22-s − 0.661·23-s − 0.931·25-s − 0.554·26-s + 0.912·28-s + 0.108·29-s − 0.148·31-s − 0.176·32-s − 0.478·35-s + 0.368·37-s + 1.10·38-s + 0.0926·40-s + 0.479·41-s + 0.252·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.425691603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.425691603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 + 0.828T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.215424480280495530001230647146, −7.81202779877343713776047510450, −7.03204847788820837429844139792, −5.98567947915903574523159720188, −5.41160910779198550703939827745, −4.47332982245831059610355813857, −3.84678193156115005921900490206, −2.39200049255737728864894148067, −1.94174142232279018629025219412, −0.71759901306785976123668095942,
0.71759901306785976123668095942, 1.94174142232279018629025219412, 2.39200049255737728864894148067, 3.84678193156115005921900490206, 4.47332982245831059610355813857, 5.41160910779198550703939827745, 5.98567947915903574523159720188, 7.03204847788820837429844139792, 7.81202779877343713776047510450, 8.215424480280495530001230647146
