L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s − 5·11-s + 4·14-s + 16-s − 2·17-s + 6·19-s − 20-s − 5·22-s + 9·23-s + 25-s + 4·28-s − 5·29-s + 3·31-s + 32-s − 2·34-s − 4·35-s + 5·37-s + 6·38-s − 40-s − 2·41-s − 7·43-s − 5·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.50·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.223·20-s − 1.06·22-s + 1.87·23-s + 1/5·25-s + 0.755·28-s − 0.928·29-s + 0.538·31-s + 0.176·32-s − 0.342·34-s − 0.676·35-s + 0.821·37-s + 0.973·38-s − 0.158·40-s − 0.312·41-s − 1.06·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.891414669\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.891414669\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.83264834850650, −15.34270881562684, −14.93398295857367, −14.51982868172206, −13.58375436832218, −13.44810811812387, −12.78295903043591, −12.02945800881518, −11.52428841388719, −11.07170477027213, −10.67016917089619, −9.936173915420598, −9.024805820719334, −8.438911270942159, −7.701309663024329, −7.480657117628339, −6.797668901051893, −5.673138685170485, −5.234048491061690, −4.806871576557714, −4.162843578517969, −3.158801942787603, −2.642838381950683, −1.718535751580558, −0.7871941586241400,
0.7871941586241400, 1.718535751580558, 2.642838381950683, 3.158801942787603, 4.162843578517969, 4.806871576557714, 5.234048491061690, 5.673138685170485, 6.797668901051893, 7.480657117628339, 7.701309663024329, 8.438911270942159, 9.024805820719334, 9.936173915420598, 10.67016917089619, 11.07170477027213, 11.52428841388719, 12.02945800881518, 12.78295903043591, 13.44810811812387, 13.58375436832218, 14.51982868172206, 14.93398295857367, 15.34270881562684, 15.83264834850650