| L(s) = 1 | − 7.48·3-s + 7.42·5-s + 21.6·7-s + 28.9·9-s + 39.4·11-s + 37.9·13-s − 55.5·15-s − 55.1·17-s + 40.6·19-s − 162.·21-s − 130.·23-s − 69.8·25-s − 14.7·27-s − 29·29-s − 279.·31-s − 294.·33-s + 161.·35-s − 226.·37-s − 284.·39-s − 77.8·41-s − 10.5·43-s + 215.·45-s + 242.·47-s + 127.·49-s + 412.·51-s − 584.·53-s + 292.·55-s + ⋯ |
| L(s) = 1 | − 1.43·3-s + 0.664·5-s + 1.17·7-s + 1.07·9-s + 1.08·11-s + 0.810·13-s − 0.956·15-s − 0.787·17-s + 0.490·19-s − 1.68·21-s − 1.18·23-s − 0.558·25-s − 0.105·27-s − 0.185·29-s − 1.62·31-s − 1.55·33-s + 0.778·35-s − 1.00·37-s − 1.16·39-s − 0.296·41-s − 0.0372·43-s + 0.713·45-s + 0.752·47-s + 0.371·49-s + 1.13·51-s − 1.51·53-s + 0.718·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 7.48T + 27T^{2} \) |
| 5 | \( 1 - 7.42T + 125T^{2} \) |
| 7 | \( 1 - 21.6T + 343T^{2} \) |
| 11 | \( 1 - 39.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 55.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 226.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 77.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 10.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 242.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 584.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 87.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 48.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 100.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 803.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 70.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 823.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.579877427947720708346330151401, −7.55872788974902670352848254579, −6.64695669279903701318710861632, −5.97485315720401263617790494554, −5.42145971108154611374790584487, −4.55605652660964079707745639428, −3.71987980221606437145491593393, −1.89655021410907941730271878142, −1.34930504802099964185116623612, 0,
1.34930504802099964185116623612, 1.89655021410907941730271878142, 3.71987980221606437145491593393, 4.55605652660964079707745639428, 5.42145971108154611374790584487, 5.97485315720401263617790494554, 6.64695669279903701318710861632, 7.55872788974902670352848254579, 8.579877427947720708346330151401
