L(s) = 1 | + 1.67·2-s + 2.07·3-s + 0.820·4-s + 1.54·5-s + 3.48·6-s + 0.0416·7-s − 1.98·8-s + 1.29·9-s + 2.59·10-s − 0.546·11-s + 1.70·12-s + 4.01·13-s + 0.0699·14-s + 3.20·15-s − 4.96·16-s + 2.09·17-s + 2.17·18-s + 7.55·19-s + 1.26·20-s + 0.0863·21-s − 0.917·22-s + 7.78·23-s − 4.10·24-s − 2.60·25-s + 6.74·26-s − 3.53·27-s + 0.0341·28-s + ⋯ |
L(s) = 1 | + 1.18·2-s + 1.19·3-s + 0.410·4-s + 0.691·5-s + 1.42·6-s + 0.0157·7-s − 0.700·8-s + 0.432·9-s + 0.821·10-s − 0.164·11-s + 0.490·12-s + 1.11·13-s + 0.0186·14-s + 0.827·15-s − 1.24·16-s + 0.508·17-s + 0.513·18-s + 1.73·19-s + 0.283·20-s + 0.0188·21-s − 0.195·22-s + 1.62·23-s − 0.838·24-s − 0.521·25-s + 1.32·26-s − 0.679·27-s + 0.00645·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.595316185\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.595316185\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2671 | \( 1 - T \) |
good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 3 | \( 1 - 2.07T + 3T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 - 0.0416T + 7T^{2} \) |
| 11 | \( 1 + 0.546T + 11T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 - 7.55T + 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 - 4.37T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 - 5.71T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 + 6.18T + 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 - 0.0854T + 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 - 6.98T + 71T^{2} \) |
| 73 | \( 1 + 7.44T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
| 97 | \( 1 + 3.54T + 97T^{2} \) |
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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
−9.042077251108210560915473929609, −8.082238662916731678079771693016, −7.35186849972162643255390921950, −6.29042341031621853083959948896, −5.61103564202452745204937200031, −4.96984658103352120688746763058, −3.81867673868707092545783160962, −3.24075261965139839682920603290, −2.60548308214549507173441092210, −1.34302521965891774808465133442,
1.34302521965891774808465133442, 2.60548308214549507173441092210, 3.24075261965139839682920603290, 3.81867673868707092545783160962, 4.96984658103352120688746763058, 5.61103564202452745204937200031, 6.29042341031621853083959948896, 7.35186849972162643255390921950, 8.082238662916731678079771693016, 9.042077251108210560915473929609