L(s) = 1 | + 2.26·2-s + 3.12·4-s + 3.26·5-s + 2.33·7-s + 2.55·8-s + 7.39·10-s − 1.17·11-s − 0.357·13-s + 5.28·14-s − 0.463·16-s + 1.97·17-s − 4.92·19-s + 10.2·20-s − 2.65·22-s + 0.133·23-s + 5.65·25-s − 0.809·26-s + 7.30·28-s − 6.96·29-s + 2.15·31-s − 6.16·32-s + 4.47·34-s + 7.61·35-s − 1.96·37-s − 11.1·38-s + 8.35·40-s − 4.35·41-s + ⋯ |
L(s) = 1 | + 1.60·2-s + 1.56·4-s + 1.45·5-s + 0.881·7-s + 0.904·8-s + 2.33·10-s − 0.352·11-s − 0.0990·13-s + 1.41·14-s − 0.115·16-s + 0.478·17-s − 1.12·19-s + 2.28·20-s − 0.565·22-s + 0.0277·23-s + 1.13·25-s − 0.158·26-s + 1.37·28-s − 1.29·29-s + 0.387·31-s − 1.09·32-s + 0.767·34-s + 1.28·35-s − 0.322·37-s − 1.80·38-s + 1.32·40-s − 0.679·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.315265055\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.315265055\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 5 | \( 1 - 3.26T + 5T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 + 0.357T + 13T^{2} \) |
| 17 | \( 1 - 1.97T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 23 | \( 1 - 0.133T + 23T^{2} \) |
| 29 | \( 1 + 6.96T + 29T^{2} \) |
| 31 | \( 1 - 2.15T + 31T^{2} \) |
| 37 | \( 1 + 1.96T + 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 - 2.67T + 43T^{2} \) |
| 47 | \( 1 + 0.574T + 47T^{2} \) |
| 53 | \( 1 - 5.50T + 53T^{2} \) |
| 59 | \( 1 + 2.14T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 8.39T + 71T^{2} \) |
| 73 | \( 1 + 1.93T + 73T^{2} \) |
| 79 | \( 1 - 1.22T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 6.22T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−9.749470268771911946191574629569, −8.848641413074943689368170714805, −7.80565919987854820763044390360, −6.73022190770756657598605502001, −6.00634383696216486225669373192, −5.32471102271559015370834381848, −4.74664906285404295120589553151, −3.65779909515882304025550318489, −2.44806312612083975975851724764, −1.77662241939864581036140030411,
1.77662241939864581036140030411, 2.44806312612083975975851724764, 3.65779909515882304025550318489, 4.74664906285404295120589553151, 5.32471102271559015370834381848, 6.00634383696216486225669373192, 6.73022190770756657598605502001, 7.80565919987854820763044390360, 8.848641413074943689368170714805, 9.749470268771911946191574629569