| L(s) = 1 | + 2-s − 1.56·3-s + 4-s + 1.51·5-s − 1.56·6-s + 2.72·7-s + 8-s − 0.560·9-s + 1.51·10-s − 1.08·11-s − 1.56·12-s − 3.25·13-s + 2.72·14-s − 2.36·15-s + 16-s − 6.54·17-s − 0.560·18-s + 7.89·19-s + 1.51·20-s − 4.25·21-s − 1.08·22-s + 23-s − 1.56·24-s − 2.71·25-s − 3.25·26-s + 5.56·27-s + 2.72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.901·3-s + 0.5·4-s + 0.676·5-s − 0.637·6-s + 1.03·7-s + 0.353·8-s − 0.186·9-s + 0.478·10-s − 0.327·11-s − 0.450·12-s − 0.901·13-s + 0.728·14-s − 0.610·15-s + 0.250·16-s − 1.58·17-s − 0.132·18-s + 1.81·19-s + 0.338·20-s − 0.928·21-s − 0.231·22-s + 0.208·23-s − 0.318·24-s − 0.542·25-s − 0.637·26-s + 1.07·27-s + 0.515·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.620662529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.620662529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 - 7.89T + 19T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 5.71T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 8.07T + 59T^{2} \) |
| 61 | \( 1 - 5.67T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 1.75T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 6.02T + 97T^{2} \) |
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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
−7.85068207179561626797705714092, −7.23262115038810650483327469999, −6.46682571468875811533082255204, −5.75534203747536256559110085172, −5.06806589833126526098582719139, −4.90857222712862462452541091014, −3.82670930767849991531896184418, −2.60953366410438022744877430675, −2.05090657598002653718618536628, −0.794400819217106577421898979854,
0.794400819217106577421898979854, 2.05090657598002653718618536628, 2.60953366410438022744877430675, 3.82670930767849991531896184418, 4.90857222712862462452541091014, 5.06806589833126526098582719139, 5.75534203747536256559110085172, 6.46682571468875811533082255204, 7.23262115038810650483327469999, 7.85068207179561626797705714092
