| L(s) = 1 | + 2-s − 4-s + 3.37·5-s + 3.69·7-s − 3·8-s + 3.37·10-s + 5.22·11-s − 4.24·13-s + 3.69·14-s − 16-s + 2.82·19-s − 3.37·20-s + 5.22·22-s − 0.634·23-s + 6.41·25-s − 4.24·26-s − 3.69·28-s + 4.46·29-s − 1.53·31-s + 5·32-s + 12.4·35-s + 2.48·37-s + 2.82·38-s − 10.1·40-s + 4.46·41-s − 5.65·43-s − 5.22·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.5·4-s + 1.51·5-s + 1.39·7-s − 1.06·8-s + 1.06·10-s + 1.57·11-s − 1.17·13-s + 0.987·14-s − 0.250·16-s + 0.648·19-s − 0.755·20-s + 1.11·22-s − 0.132·23-s + 1.28·25-s − 0.832·26-s − 0.698·28-s + 0.828·29-s − 0.274·31-s + 0.883·32-s + 2.11·35-s + 0.408·37-s + 0.458·38-s − 1.60·40-s + 0.696·41-s − 0.862·43-s − 0.787·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.578844636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.578844636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 - 5.22T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 0.634T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 + 1.53T + 31T^{2} \) |
| 37 | \( 1 - 2.48T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 0.242T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 - 0.634T + 71T^{2} \) |
| 73 | \( 1 - 9.23T + 73T^{2} \) |
| 79 | \( 1 + 8.02T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 9.68T + 97T^{2} \) |
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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
−9.059092889293690449178236038291, −8.238997566739497277557251262713, −7.21697017372188350658957359343, −6.27812532141883533828071485175, −5.67532422556965809895655855280, −4.83006377224843928807455738385, −4.46961446154228375015318954889, −3.20301746708500458303548955190, −2.09786209232131405592120585234, −1.21772643226426224353855559022,
1.21772643226426224353855559022, 2.09786209232131405592120585234, 3.20301746708500458303548955190, 4.46961446154228375015318954889, 4.83006377224843928807455738385, 5.67532422556965809895655855280, 6.27812532141883533828071485175, 7.21697017372188350658957359343, 8.238997566739497277557251262713, 9.059092889293690449178236038291
