| L(s) = 1 | − 1.21·2-s − 0.529·4-s + 5-s − 3.37·7-s + 3.06·8-s − 1.21·10-s + 11-s − 0.439·13-s + 4.08·14-s − 2.65·16-s + 3.08·17-s − 19-s − 0.529·20-s − 1.21·22-s − 1.45·23-s + 25-s + 0.533·26-s + 1.78·28-s − 5.85·29-s − 6.80·31-s − 2.91·32-s − 3.74·34-s − 3.37·35-s − 2.73·37-s + 1.21·38-s + 3.06·40-s + 3.67·41-s + ⋯ |
| L(s) = 1 | − 0.857·2-s − 0.264·4-s + 0.447·5-s − 1.27·7-s + 1.08·8-s − 0.383·10-s + 0.301·11-s − 0.121·13-s + 1.09·14-s − 0.664·16-s + 0.749·17-s − 0.229·19-s − 0.118·20-s − 0.258·22-s − 0.303·23-s + 0.200·25-s + 0.104·26-s + 0.337·28-s − 1.08·29-s − 1.22·31-s − 0.514·32-s − 0.642·34-s − 0.569·35-s − 0.449·37-s + 0.196·38-s + 0.485·40-s + 0.574·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7150552497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7150552497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 13 | \( 1 + 0.439T + 13T^{2} \) |
| 17 | \( 1 - 3.08T + 17T^{2} \) |
| 23 | \( 1 + 1.45T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 + 6.80T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 + 0.210T + 47T^{2} \) |
| 53 | \( 1 + 5.17T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.66T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 - 7.15T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 3.35T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−7.75792493057507503658186945928, −7.11888364710085971344184992146, −6.47439857822569900169453640706, −5.69003542582746688160579092019, −5.10818434330440949585560706941, −3.98014218486112232620661044526, −3.54531501006253685920484117386, −2.44695661425238986924816461826, −1.54059190570268608819639552089, −0.47597432122503344972523003547,
0.47597432122503344972523003547, 1.54059190570268608819639552089, 2.44695661425238986924816461826, 3.54531501006253685920484117386, 3.98014218486112232620661044526, 5.10818434330440949585560706941, 5.69003542582746688160579092019, 6.47439857822569900169453640706, 7.11888364710085971344184992146, 7.75792493057507503658186945928
