| L(s) = 1 | − 10.3·3-s + 14.1·5-s − 4.06·7-s + 79.2·9-s − 5.53·11-s − 45.6·13-s − 145.·15-s − 92.8·17-s + 19·19-s + 41.8·21-s − 57.0·23-s + 75.1·25-s − 538.·27-s − 290.·29-s + 163.·31-s + 57.0·33-s − 57.4·35-s − 81.3·37-s + 470.·39-s + 73.1·41-s − 346.·43-s + 1.12e3·45-s + 503.·47-s − 326.·49-s + 957.·51-s − 164.·53-s − 78.2·55-s + ⋯ |
| L(s) = 1 | − 1.98·3-s + 1.26·5-s − 0.219·7-s + 2.93·9-s − 0.151·11-s − 0.974·13-s − 2.51·15-s − 1.32·17-s + 0.229·19-s + 0.435·21-s − 0.517·23-s + 0.601·25-s − 3.83·27-s − 1.85·29-s + 0.947·31-s + 0.300·33-s − 0.277·35-s − 0.361·37-s + 1.93·39-s + 0.278·41-s − 1.22·43-s + 3.71·45-s + 1.56·47-s − 0.951·49-s + 2.62·51-s − 0.425·53-s − 0.191·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 10.3T + 27T^{2} \) |
| 5 | \( 1 - 14.1T + 125T^{2} \) |
| 7 | \( 1 + 4.06T + 343T^{2} \) |
| 11 | \( 1 + 5.53T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.8T + 4.91e3T^{2} \) |
| 23 | \( 1 + 57.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 290.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 163.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 81.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 73.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 503.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 164.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 763.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 545.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 510.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 294.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 172.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 973.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 510.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 845.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 863.T + 9.12e5T^{2} \) |
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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
−11.95980899150039081021207445525, −11.00781401704267255453617188635, −10.12353048500049309578917150465, −9.433592377329433192821988870356, −7.27235713277545819774648647338, −6.31437177828056696038683492349, −5.54277737509315413162524977558, −4.54572926229080304835098145911, −1.86395500062007083049934331958, 0,
1.86395500062007083049934331958, 4.54572926229080304835098145911, 5.54277737509315413162524977558, 6.31437177828056696038683492349, 7.27235713277545819774648647338, 9.433592377329433192821988870356, 10.12353048500049309578917150465, 11.00781401704267255453617188635, 11.95980899150039081021207445525
