| L(s) = 1 | − 2.78·2-s + 3.17·3-s + 5.74·4-s − 8.84·6-s − 7-s − 10.4·8-s + 7.09·9-s − 11-s + 18.2·12-s + 0.540·13-s + 2.78·14-s + 17.5·16-s + 2.54·17-s − 19.7·18-s + 8.20·19-s − 3.17·21-s + 2.78·22-s + 4.34·23-s − 33.1·24-s − 1.50·26-s + 13.0·27-s − 5.74·28-s − 3.40·29-s + 1.01·31-s − 27.8·32-s − 3.17·33-s − 7.08·34-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 1.83·3-s + 2.87·4-s − 3.60·6-s − 0.377·7-s − 3.68·8-s + 2.36·9-s − 0.301·11-s + 5.26·12-s + 0.149·13-s + 0.743·14-s + 4.37·16-s + 0.617·17-s − 4.65·18-s + 1.88·19-s − 0.693·21-s + 0.593·22-s + 0.905·23-s − 6.75·24-s − 0.294·26-s + 2.50·27-s − 1.08·28-s − 0.632·29-s + 0.181·31-s − 4.92·32-s − 0.553·33-s − 1.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520831591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520831591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 - 3.17T + 3T^{2} \) |
| 13 | \( 1 - 0.540T + 13T^{2} \) |
| 17 | \( 1 - 2.54T + 17T^{2} \) |
| 19 | \( 1 - 8.20T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 + 3.40T + 29T^{2} \) |
| 31 | \( 1 - 1.01T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 + 8.84T + 47T^{2} \) |
| 53 | \( 1 - 2.77T + 53T^{2} \) |
| 59 | \( 1 + 0.0440T + 59T^{2} \) |
| 61 | \( 1 - 5.52T + 61T^{2} \) |
| 67 | \( 1 - 3.03T + 67T^{2} \) |
| 71 | \( 1 + 2.69T + 71T^{2} \) |
| 73 | \( 1 - 9.20T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 + 8.06T + 83T^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 - 8.21T + 97T^{2} \) |
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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
−9.219036104396366324687015986835, −8.458508126064569396471036209140, −7.929439041980905743504258344134, −7.28099324591628845268465946958, −6.74716410702606733094326271613, −5.38037135620749264468749164254, −3.41139540570332174789718017006, −3.08771499822736423262408841614, −2.03301220536844196756348098555, −1.07696084164497808710931597246,
1.07696084164497808710931597246, 2.03301220536844196756348098555, 3.08771499822736423262408841614, 3.41139540570332174789718017006, 5.38037135620749264468749164254, 6.74716410702606733094326271613, 7.28099324591628845268465946958, 7.929439041980905743504258344134, 8.458508126064569396471036209140, 9.219036104396366324687015986835
