L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s + 11-s + 12-s + 5·13-s − 14-s + 3·15-s + 16-s + 3·17-s − 18-s − 2·19-s + 3·20-s + 21-s − 22-s − 24-s + 4·25-s − 5·26-s + 27-s + 28-s − 3·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.301·11-s + 0.288·12-s + 1.38·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.458·19-s + 0.670·20-s + 0.218·21-s − 0.213·22-s − 0.204·24-s + 4/5·25-s − 0.980·26-s + 0.192·27-s + 0.188·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.876557270\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.876557270\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.76414459321098, −14.50064820944557, −13.94242324747256, −13.40201671266762, −12.99451703833762, −12.41326601993683, −11.62207864225262, −11.10278910735374, −10.56463084649911, −10.09306266325483, −9.471568116972639, −9.146174802869908, −8.549621975857137, −8.091245774014620, −7.462857704391833, −6.728870248675268, −6.198857077439798, −5.712865288141470, −5.113149658276519, −4.093546481567118, −3.559328130968322, −2.710354866105521, −2.044195809749019, −1.496501024109554, −0.8471832490812269,
0.8471832490812269, 1.496501024109554, 2.044195809749019, 2.710354866105521, 3.559328130968322, 4.093546481567118, 5.113149658276519, 5.712865288141470, 6.198857077439798, 6.728870248675268, 7.462857704391833, 8.091245774014620, 8.549621975857137, 9.146174802869908, 9.471568116972639, 10.09306266325483, 10.56463084649911, 11.10278910735374, 11.62207864225262, 12.41326601993683, 12.99451703833762, 13.40201671266762, 13.94242324747256, 14.50064820944557, 14.76414459321098