L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 7-s − 2·9-s − 2·10-s + 5·11-s + 2·12-s + 2·14-s − 15-s − 4·16-s + 2·17-s − 4·18-s + 6·19-s − 2·20-s + 21-s + 10·22-s − 6·23-s + 25-s − 5·27-s + 2·28-s + 2·29-s − 2·30-s + 8·31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s − 2/3·9-s − 0.632·10-s + 1.50·11-s + 0.577·12-s + 0.534·14-s − 0.258·15-s − 16-s + 0.485·17-s − 0.942·18-s + 1.37·19-s − 0.447·20-s + 0.218·21-s + 2.13·22-s − 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.377·28-s + 0.371·29-s − 0.365·30-s + 1.43·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.328663992\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.328663992\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.968225307166566464300665819551, −7.38582777598806693551933482738, −6.27950698879704291894289728948, −6.02437081425343730884933598968, −5.04041858480608539475486444949, −4.33320241614309115109722075816, −3.70908410568190611935364437247, −3.10021973870422809922034328206, −2.28450753048798176281985506072, −0.991569683651555248883208150594,
0.991569683651555248883208150594, 2.28450753048798176281985506072, 3.10021973870422809922034328206, 3.70908410568190611935364437247, 4.33320241614309115109722075816, 5.04041858480608539475486444949, 6.02437081425343730884933598968, 6.27950698879704291894289728948, 7.38582777598806693551933482738, 7.968225307166566464300665819551