| L(s) = 1 | − 5·5-s + 1.21·7-s − 33.6·11-s + 77.5·13-s − 73.2·17-s + 39.2·19-s − 35.2·23-s + 25·25-s + 15.7·29-s + 52.3·31-s − 6.06·35-s − 67.1·37-s + 25.0·41-s − 31.7·43-s + 412.·47-s − 341.·49-s − 54.7·53-s + 168.·55-s − 515.·59-s + 862.·61-s − 387.·65-s − 367.·67-s + 930.·71-s − 797.·73-s − 40.7·77-s − 85.5·79-s − 91.9·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.0654·7-s − 0.921·11-s + 1.65·13-s − 1.04·17-s + 0.473·19-s − 0.319·23-s + 0.200·25-s + 0.101·29-s + 0.303·31-s − 0.0292·35-s − 0.298·37-s + 0.0953·41-s − 0.112·43-s + 1.27·47-s − 0.995·49-s − 0.141·53-s + 0.412·55-s − 1.13·59-s + 1.81·61-s − 0.740·65-s − 0.670·67-s + 1.55·71-s − 1.27·73-s − 0.0603·77-s − 0.121·79-s − 0.121·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 1.21T + 343T^{2} \) |
| 11 | \( 1 + 33.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 35.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 15.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 52.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 67.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 25.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 31.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 412.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 54.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 515.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 862.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 797.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 85.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 91.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 888.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 333.T + 9.12e5T^{2} \) |
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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
−8.504876514328364478120089849612, −7.998330555485162549997441855577, −7.03567817588727025687454947278, −6.21210259937142090103756390795, −5.35807585071460929614729146835, −4.36870472057222436944668857329, −3.54666514362136180679345923488, −2.51706589080580863077742100334, −1.25260398047175593072894494025, 0,
1.25260398047175593072894494025, 2.51706589080580863077742100334, 3.54666514362136180679345923488, 4.36870472057222436944668857329, 5.35807585071460929614729146835, 6.21210259937142090103756390795, 7.03567817588727025687454947278, 7.998330555485162549997441855577, 8.504876514328364478120089849612
