| L(s) = 1 | + 21.5·3-s + 14.7·5-s + 222.·9-s + 58.5·11-s − 1.17e3·13-s + 317.·15-s − 1.49e3·17-s − 498.·19-s − 1.88e3·23-s − 2.90e3·25-s − 443.·27-s + 1.91e3·29-s − 794.·31-s + 1.26e3·33-s + 2.98e3·37-s − 2.54e4·39-s − 1.19e4·41-s + 9.82e3·43-s + 3.27e3·45-s + 1.96e4·47-s − 3.22e4·51-s − 1.98e4·53-s + 862.·55-s − 1.07e4·57-s − 3.58e4·59-s − 4.99e4·61-s − 1.73e4·65-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 0.263·5-s + 0.915·9-s + 0.145·11-s − 1.93·13-s + 0.364·15-s − 1.25·17-s − 0.317·19-s − 0.744·23-s − 0.930·25-s − 0.117·27-s + 0.422·29-s − 0.148·31-s + 0.201·33-s + 0.358·37-s − 2.67·39-s − 1.10·41-s + 0.809·43-s + 0.241·45-s + 1.29·47-s − 1.73·51-s − 0.971·53-s + 0.0384·55-s − 0.438·57-s − 1.34·59-s − 1.71·61-s − 0.509·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 21.5T + 243T^{2} \) |
| 5 | \( 1 - 14.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 58.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.49e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 498.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.88e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 794.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.98e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.98e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.35e4T + 8.58e9T^{2} \) |
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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
−9.661230428671805496926848190197, −9.258308195658012423100159337647, −8.164168145560100094097554930807, −7.47072399774645150014467495255, −6.38499738990102780412054354784, −4.92452096408087648170908803931, −3.90274727806723163269633803715, −2.58164548455929411027220138504, −2.00844090204720046645167753416, 0,
2.00844090204720046645167753416, 2.58164548455929411027220138504, 3.90274727806723163269633803715, 4.92452096408087648170908803931, 6.38499738990102780412054354784, 7.47072399774645150014467495255, 8.164168145560100094097554930807, 9.258308195658012423100159337647, 9.661230428671805496926848190197
