| L(s) = 1 | − 2-s + 4-s + 2.24·5-s − 0.679·7-s − 8-s − 2.24·10-s + 0.775·11-s − 3.33·13-s + 0.679·14-s + 16-s − 2.56·17-s + 0.994·19-s + 2.24·20-s − 0.775·22-s − 23-s + 0.0521·25-s + 3.33·26-s − 0.679·28-s − 3.77·29-s − 1.26·31-s − 32-s + 2.56·34-s − 1.52·35-s + 8.73·37-s − 0.994·38-s − 2.24·40-s − 0.329·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.00·5-s − 0.256·7-s − 0.353·8-s − 0.710·10-s + 0.233·11-s − 0.925·13-s + 0.181·14-s + 0.250·16-s − 0.622·17-s + 0.228·19-s + 0.502·20-s − 0.165·22-s − 0.208·23-s + 0.0104·25-s + 0.654·26-s − 0.128·28-s − 0.701·29-s − 0.226·31-s − 0.176·32-s + 0.440·34-s − 0.258·35-s + 1.43·37-s − 0.161·38-s − 0.355·40-s − 0.0515·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2.24T + 5T^{2} \) |
| 7 | \( 1 + 0.679T + 7T^{2} \) |
| 11 | \( 1 - 0.775T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 0.994T + 19T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 8.73T + 37T^{2} \) |
| 41 | \( 1 + 0.329T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.86T + 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 + 0.571T + 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 - 1.53T + 71T^{2} \) |
| 73 | \( 1 - 0.859T + 73T^{2} \) |
| 79 | \( 1 - 0.596T + 79T^{2} \) |
| 83 | \( 1 - 0.367T + 83T^{2} \) |
| 89 | \( 1 + 9.85T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.131016091156696745848482833618, −7.51333760694654130658901320897, −6.62475476293348826972180179095, −6.11849165113948282377051034938, −5.27914774910349315496791904401, −4.37713142262178371885503318307, −3.16198548057716173748652172151, −2.29460611674173441731845690618, −1.50752331715854792550861665025, 0,
1.50752331715854792550861665025, 2.29460611674173441731845690618, 3.16198548057716173748652172151, 4.37713142262178371885503318307, 5.27914774910349315496791904401, 6.11849165113948282377051034938, 6.62475476293348826972180179095, 7.51333760694654130658901320897, 8.131016091156696745848482833618
