| L(s) = 1 | + 2.67·2-s + 5.17·4-s + 0.923·5-s − 3.83·7-s + 8.49·8-s + 2.47·10-s − 5.25·11-s − 2.39·13-s − 10.2·14-s + 12.4·16-s − 1.49·17-s − 2.22·19-s + 4.77·20-s − 14.0·22-s − 0.388·23-s − 4.14·25-s − 6.40·26-s − 19.8·28-s − 2.74·31-s + 16.2·32-s − 4.00·34-s − 3.54·35-s + 10.6·37-s − 5.95·38-s + 7.85·40-s − 6.13·41-s + 0.0976·43-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 2.58·4-s + 0.413·5-s − 1.44·7-s + 3.00·8-s + 0.782·10-s − 1.58·11-s − 0.663·13-s − 2.74·14-s + 3.10·16-s − 0.363·17-s − 0.510·19-s + 1.06·20-s − 3.00·22-s − 0.0810·23-s − 0.829·25-s − 1.25·26-s − 3.74·28-s − 0.493·31-s + 2.87·32-s − 0.687·34-s − 0.598·35-s + 1.75·37-s − 0.965·38-s + 1.24·40-s − 0.957·41-s + 0.0148·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 - 0.923T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 + 0.388T + 23T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 6.13T + 41T^{2} \) |
| 43 | \( 1 - 0.0976T + 43T^{2} \) |
| 47 | \( 1 + 2.69T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 3.44T + 59T^{2} \) |
| 61 | \( 1 + 8.67T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 3.13T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 6.57T + 89T^{2} \) |
| 97 | \( 1 - 9.57T + 97T^{2} \) |
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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
−7.29142435934543714791704677430, −6.51801866640105981564983203554, −5.93942754056067418509554096546, −5.50194741954554523589214212914, −4.67176285523529536995713955789, −4.03551819844432021361592020906, −3.05169487859090120969482087711, −2.69636376442322949439874062874, −1.92219133129563875262464111687, 0,
1.92219133129563875262464111687, 2.69636376442322949439874062874, 3.05169487859090120969482087711, 4.03551819844432021361592020906, 4.67176285523529536995713955789, 5.50194741954554523589214212914, 5.93942754056067418509554096546, 6.51801866640105981564983203554, 7.29142435934543714791704677430
