L(s) = 1 | + 2-s + 2.24·3-s + 4-s − 5-s + 2.24·6-s − 3.06·7-s + 8-s + 2.06·9-s − 10-s + 0.660·11-s + 2.24·12-s − 4.57·13-s − 3.06·14-s − 2.24·15-s + 16-s + 5.11·17-s + 2.06·18-s + 1.52·19-s − 20-s − 6.90·21-s + 0.660·22-s − 7.85·23-s + 2.24·24-s + 25-s − 4.57·26-s − 2.11·27-s − 3.06·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.447·5-s + 0.918·6-s − 1.15·7-s + 0.353·8-s + 0.687·9-s − 0.316·10-s + 0.199·11-s + 0.649·12-s − 1.26·13-s − 0.820·14-s − 0.580·15-s + 0.250·16-s + 1.24·17-s + 0.485·18-s + 0.350·19-s − 0.223·20-s − 1.50·21-s + 0.140·22-s − 1.63·23-s + 0.459·24-s + 0.200·25-s − 0.896·26-s − 0.406·27-s − 0.579·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 7 | \( 1 + 3.06T + 7T^{2} \) |
| 11 | \( 1 - 0.660T + 11T^{2} \) |
| 13 | \( 1 + 4.57T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 - 1.52T + 19T^{2} \) |
| 23 | \( 1 + 7.85T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 + 5.72T + 31T^{2} \) |
| 37 | \( 1 + 7.35T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 8.81T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 1.43T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 0.340T + 79T^{2} \) |
| 83 | \( 1 - 7.38T + 83T^{2} \) |
| 89 | \( 1 - 2.35T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−7.49250226669029737797701074182, −7.35583127728270906047368880598, −6.25421433346523950716666651508, −5.59957967458773502737583923431, −4.64577283275683765431609172807, −3.64919493496203347213857131720, −3.41873315230970220259829153034, −2.61953616345193984587665221244, −1.76838426814166684209408256078, 0,
1.76838426814166684209408256078, 2.61953616345193984587665221244, 3.41873315230970220259829153034, 3.64919493496203347213857131720, 4.64577283275683765431609172807, 5.59957967458773502737583923431, 6.25421433346523950716666651508, 7.35583127728270906047368880598, 7.49250226669029737797701074182