| L(s) = 1 | − 2.37·2-s − 6.00·3-s − 2.37·4-s + 22.0·5-s + 14.2·6-s + 21.9·7-s + 24.6·8-s + 9.11·9-s − 52.1·10-s + 11·11-s + 14.2·12-s − 51.9·14-s − 132.·15-s − 39.3·16-s − 45.0·17-s − 21.6·18-s − 118.·19-s − 52.2·20-s − 131.·21-s − 26.0·22-s + 37.3·23-s − 147.·24-s + 359.·25-s + 107.·27-s − 52.0·28-s − 137.·29-s + 313.·30-s + ⋯ |
| L(s) = 1 | − 0.838·2-s − 1.15·3-s − 0.296·4-s + 1.96·5-s + 0.969·6-s + 1.18·7-s + 1.08·8-s + 0.337·9-s − 1.65·10-s + 0.301·11-s + 0.343·12-s − 0.992·14-s − 2.27·15-s − 0.615·16-s − 0.642·17-s − 0.283·18-s − 1.43·19-s − 0.584·20-s − 1.36·21-s − 0.252·22-s + 0.338·23-s − 1.25·24-s + 2.87·25-s + 0.766·27-s − 0.351·28-s − 0.880·29-s + 1.90·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.319303028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319303028\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.37T + 8T^{2} \) |
| 3 | \( 1 + 6.00T + 27T^{2} \) |
| 5 | \( 1 - 22.0T + 125T^{2} \) |
| 7 | \( 1 - 21.9T + 343T^{2} \) |
| 17 | \( 1 + 45.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 37.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 1.53T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 72.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 60.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 0.405T + 1.48e5T^{2} \) |
| 59 | \( 1 + 37.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 427.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 186.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 831.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 840.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.23e3T + 9.12e5T^{2} \) |
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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
−8.852739129221344370310503950815, −8.486435997414790093214579689927, −7.20865507686584903422232648228, −6.38166018131668011937590437780, −5.69045626675517914857428680252, −4.99995902041839080047718893319, −4.35614461264906527425345109328, −2.29972798541246092824061129282, −1.60761765035166993994556681094, −0.69090340069554749819069713240,
0.69090340069554749819069713240, 1.60761765035166993994556681094, 2.29972798541246092824061129282, 4.35614461264906527425345109328, 4.99995902041839080047718893319, 5.69045626675517914857428680252, 6.38166018131668011937590437780, 7.20865507686584903422232648228, 8.486435997414790093214579689927, 8.852739129221344370310503950815
