L(s) = 1 | − 2-s − 1.00·3-s + 4-s − 1.06·5-s + 1.00·6-s + 0.645·7-s − 8-s − 1.99·9-s + 1.06·10-s − 0.354·11-s − 1.00·12-s − 2.49·13-s − 0.645·14-s + 1.06·15-s + 16-s − 3.08·17-s + 1.99·18-s + 19-s − 1.06·20-s − 0.648·21-s + 0.354·22-s − 6.41·23-s + 1.00·24-s − 3.87·25-s + 2.49·26-s + 5.01·27-s + 0.645·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.580·3-s + 0.5·4-s − 0.474·5-s + 0.410·6-s + 0.243·7-s − 0.353·8-s − 0.663·9-s + 0.335·10-s − 0.106·11-s − 0.290·12-s − 0.693·13-s − 0.172·14-s + 0.275·15-s + 0.250·16-s − 0.747·17-s + 0.469·18-s + 0.229·19-s − 0.237·20-s − 0.141·21-s + 0.0756·22-s − 1.33·23-s + 0.205·24-s − 0.774·25-s + 0.490·26-s + 0.964·27-s + 0.121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3243236574\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3243236574\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 1.00T + 3T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 - 0.645T + 7T^{2} \) |
| 11 | \( 1 + 0.354T + 11T^{2} \) |
| 13 | \( 1 + 2.49T + 13T^{2} \) |
| 17 | \( 1 + 3.08T + 17T^{2} \) |
| 23 | \( 1 + 6.41T + 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 2.04T + 41T^{2} \) |
| 43 | \( 1 + 6.64T + 43T^{2} \) |
| 47 | \( 1 + 2.89T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 3.05T + 61T^{2} \) |
| 67 | \( 1 + 0.563T + 67T^{2} \) |
| 71 | \( 1 + 9.10T + 71T^{2} \) |
| 73 | \( 1 - 5.14T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.17T + 83T^{2} \) |
| 89 | \( 1 + 8.89T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.934439815927701727149118102335, −7.26758723541616172584106484844, −6.38750092367917455343511725111, −6.00183317833680785892612706634, −5.00545320628269205738966653867, −4.46386418977890773367636109832, −3.37657419726197202465331922662, −2.55460425045292895778981931527, −1.64098608150667609157228280125, −0.31530543217354069782237753736,
0.31530543217354069782237753736, 1.64098608150667609157228280125, 2.55460425045292895778981931527, 3.37657419726197202465331922662, 4.46386418977890773367636109832, 5.00545320628269205738966653867, 6.00183317833680785892612706634, 6.38750092367917455343511725111, 7.26758723541616172584106484844, 7.934439815927701727149118102335