| L(s) = 1 | − 4.69·3-s − 10.2·7-s − 220.·9-s + 596.·11-s + 420.·13-s − 974.·17-s − 380.·19-s + 48.1·21-s − 3.54e3·23-s + 2.17e3·27-s + 5.44e3·29-s − 3.62e3·31-s − 2.79e3·33-s − 1.75e3·37-s − 1.97e3·39-s + 263.·41-s − 1.44e4·43-s − 2.34e4·47-s − 1.67e4·49-s + 4.57e3·51-s − 3.34e4·53-s + 1.78e3·57-s − 2.90e3·59-s + 2.94e4·61-s + 2.26e3·63-s − 7.16e3·67-s + 1.66e4·69-s + ⋯ |
| L(s) = 1 | − 0.301·3-s − 0.0791·7-s − 0.909·9-s + 1.48·11-s + 0.690·13-s − 0.817·17-s − 0.241·19-s + 0.0238·21-s − 1.39·23-s + 0.574·27-s + 1.20·29-s − 0.677·31-s − 0.447·33-s − 0.210·37-s − 0.207·39-s + 0.0245·41-s − 1.18·43-s − 1.54·47-s − 0.993·49-s + 0.246·51-s − 1.63·53-s + 0.0728·57-s − 0.108·59-s + 1.01·61-s + 0.0719·63-s − 0.194·67-s + 0.420·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.69T + 243T^{2} \) |
| 7 | \( 1 + 10.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 596.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 420.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 974.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 380.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 263.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.44e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.34e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.16e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.29e4T + 8.58e9T^{2} \) |
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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
−11.35602276898815850218431250527, −10.14248156823792130540740670181, −8.988231197571559199012981444767, −8.254001113523959168202724951423, −6.63921849102028348389506345939, −6.04649126806450969750431654022, −4.55598021586551119358782022351, −3.33701990116238506136572872889, −1.64148596388471484285959430059, 0,
1.64148596388471484285959430059, 3.33701990116238506136572872889, 4.55598021586551119358782022351, 6.04649126806450969750431654022, 6.63921849102028348389506345939, 8.254001113523959168202724951423, 8.988231197571559199012981444767, 10.14248156823792130540740670181, 11.35602276898815850218431250527
