L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 5·5-s − 6·6-s + 12·7-s + 8·8-s + 9·9-s + 10·10-s + 37·11-s − 12·12-s + 19·13-s + 24·14-s − 15·15-s + 16·16-s + 17·17-s + 18·18-s + 37·19-s + 20·20-s − 36·21-s + 74·22-s − 3·23-s − 24·24-s − 100·25-s + 38·26-s − 27·27-s + 48·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.647·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.01·11-s − 0.288·12-s + 0.405·13-s + 0.458·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.446·19-s + 0.223·20-s − 0.374·21-s + 0.717·22-s − 0.0271·23-s − 0.204·24-s − 4/5·25-s + 0.286·26-s − 0.192·27-s + 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.330635747\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.330635747\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 17 | \( 1 - p T \) |
good | 5 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 37 T + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 19 | \( 1 - 37 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 T + p^{3} T^{2} \) |
| 29 | \( 1 + 86 T + p^{3} T^{2} \) |
| 31 | \( 1 + 142 T + p^{3} T^{2} \) |
| 37 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 121 T + p^{3} T^{2} \) |
| 43 | \( 1 - 3 T + p^{3} T^{2} \) |
| 47 | \( 1 - 402 T + p^{3} T^{2} \) |
| 53 | \( 1 - 174 T + p^{3} T^{2} \) |
| 59 | \( 1 - 270 T + p^{3} T^{2} \) |
| 61 | \( 1 + 520 T + p^{3} T^{2} \) |
| 67 | \( 1 + 780 T + p^{3} T^{2} \) |
| 71 | \( 1 - 84 T + p^{3} T^{2} \) |
| 73 | \( 1 + 302 T + p^{3} T^{2} \) |
| 79 | \( 1 - 178 T + p^{3} T^{2} \) |
| 83 | \( 1 - 698 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1512 T + p^{3} T^{2} \) |
| 97 | \( 1 + 500 T + p^{3} T^{2} \) |
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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
−13.44092654516132302689916159958, −12.18434491403111496489834604027, −11.46923469239353842591973267823, −10.41208178909743828330693651066, −9.071804812449913545579322418352, −7.47169879447251191436078965196, −6.21924868425965038901060308368, −5.21227854404398461206954567051, −3.80875045662764260703873870255, −1.62707210108612188764142658187,
1.62707210108612188764142658187, 3.80875045662764260703873870255, 5.21227854404398461206954567051, 6.21924868425965038901060308368, 7.47169879447251191436078965196, 9.071804812449913545579322418352, 10.41208178909743828330693651066, 11.46923469239353842591973267823, 12.18434491403111496489834604027, 13.44092654516132302689916159958