| L(s) = 1 | − 2-s − 0.602·3-s + 4-s + 0.602·6-s − 3.63·7-s − 8-s − 2.63·9-s + 2·11-s − 0.602·12-s − 13-s + 3.63·14-s + 16-s + 6.67·17-s + 2.63·18-s + 8.06·19-s + 2.19·21-s − 2·22-s − 0.794·23-s + 0.602·24-s + 26-s + 3.39·27-s − 3.63·28-s − 8.06·29-s + 5.20·31-s − 32-s − 1.20·33-s − 6.67·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.347·3-s + 0.5·4-s + 0.246·6-s − 1.37·7-s − 0.353·8-s − 0.878·9-s + 0.603·11-s − 0.173·12-s − 0.277·13-s + 0.971·14-s + 0.250·16-s + 1.61·17-s + 0.621·18-s + 1.85·19-s + 0.478·21-s − 0.426·22-s − 0.165·23-s + 0.123·24-s + 0.196·26-s + 0.653·27-s − 0.687·28-s − 1.49·29-s + 0.934·31-s − 0.176·32-s − 0.209·33-s − 1.14·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7639962416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7639962416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.602T + 3T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 - 8.06T + 19T^{2} \) |
| 23 | \( 1 + 0.794T + 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 - 5.20T + 31T^{2} \) |
| 37 | \( 1 + 0.431T + 37T^{2} \) |
| 41 | \( 1 - 6.86T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 - 7.63T + 47T^{2} \) |
| 53 | \( 1 + 0.794T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 + 2.86T + 61T^{2} \) |
| 67 | \( 1 - 5.20T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 1.93T + 79T^{2} \) |
| 83 | \( 1 + 2.06T + 83T^{2} \) |
| 89 | \( 1 + 8.06T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−10.33390430688334289831122316849, −9.591577753823822880968460143484, −9.134731700858641157000767503858, −7.85351887663922066212020501642, −7.12664628801960118285361409177, −6.02670337933847354491310509302, −5.49205000164514221989237480455, −3.63258094491979259447341770518, −2.80715986911783268604616530037, −0.844132271415937913603633628376,
0.844132271415937913603633628376, 2.80715986911783268604616530037, 3.63258094491979259447341770518, 5.49205000164514221989237480455, 6.02670337933847354491310509302, 7.12664628801960118285361409177, 7.85351887663922066212020501642, 9.134731700858641157000767503858, 9.591577753823822880968460143484, 10.33390430688334289831122316849
