| L(s) = 1 | + 2·2-s + 3-s − 28·4-s + 2·6-s + 167·7-s − 120·8-s − 242·9-s + 262·11-s − 28·12-s − 749·13-s + 334·14-s + 656·16-s + 1.59e3·17-s − 484·18-s − 361·19-s + 167·21-s + 524·22-s + 2.01e3·23-s − 120·24-s − 1.49e3·26-s − 485·27-s − 4.67e3·28-s − 1.05e3·29-s − 1.54e3·31-s + 5.15e3·32-s + 262·33-s + 3.19e3·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.0641·3-s − 7/8·4-s + 0.0226·6-s + 1.28·7-s − 0.662·8-s − 0.995·9-s + 0.652·11-s − 0.0561·12-s − 1.22·13-s + 0.455·14-s + 0.640·16-s + 1.34·17-s − 0.352·18-s − 0.229·19-s + 0.0826·21-s + 0.230·22-s + 0.792·23-s − 0.0425·24-s − 0.434·26-s − 0.128·27-s − 1.12·28-s − 0.232·29-s − 0.289·31-s + 0.889·32-s + 0.0418·33-s + 0.473·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 3 | \( 1 - T + p^{5} T^{2} \) |
| 7 | \( 1 - 167 T + p^{5} T^{2} \) |
| 11 | \( 1 - 262 T + p^{5} T^{2} \) |
| 13 | \( 1 + 749 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1597 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2011 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1055 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1548 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9378 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10248 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10544 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6912 T + p^{5} T^{2} \) |
| 53 | \( 1 - 35291 T + p^{5} T^{2} \) |
| 59 | \( 1 - 33655 T + p^{5} T^{2} \) |
| 61 | \( 1 + 26218 T + p^{5} T^{2} \) |
| 67 | \( 1 + 45083 T + p^{5} T^{2} \) |
| 71 | \( 1 - 30942 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46969 T + p^{5} T^{2} \) |
| 79 | \( 1 + 64430 T + p^{5} T^{2} \) |
| 83 | \( 1 - 13986 T + p^{5} T^{2} \) |
| 89 | \( 1 + 137700 T + p^{5} T^{2} \) |
| 97 | \( 1 - 22162 T + p^{5} T^{2} \) |
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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
−9.670905684734627528547060444753, −8.753205369208456604020132485827, −8.141775257018963165941539145239, −7.08127698135832291026214672257, −5.51540182057066176405692172621, −5.13655968901111827122366374475, −4.01570397409697175033935708711, −2.87295741138326132165976954541, −1.38341909395407993174926851457, 0,
1.38341909395407993174926851457, 2.87295741138326132165976954541, 4.01570397409697175033935708711, 5.13655968901111827122366374475, 5.51540182057066176405692172621, 7.08127698135832291026214672257, 8.141775257018963165941539145239, 8.753205369208456604020132485827, 9.670905684734627528547060444753
