| L(s) = 1 | + 3.12·2-s + 27.8·3-s − 22.2·4-s − 25·5-s + 86.9·6-s − 169.·8-s + 533.·9-s − 78.0·10-s − 405.·11-s − 620.·12-s − 931.·13-s − 696.·15-s + 184.·16-s − 736.·17-s + 1.66e3·18-s − 994.·19-s + 556.·20-s − 1.26e3·22-s + 1.22e3·23-s − 4.71e3·24-s + 625·25-s − 2.90e3·26-s + 8.10e3·27-s − 5.58e3·29-s − 2.17e3·30-s + 3.14e3·31-s + 5.99e3·32-s + ⋯ |
| L(s) = 1 | + 0.551·2-s + 1.78·3-s − 0.695·4-s − 0.447·5-s + 0.986·6-s − 0.935·8-s + 2.19·9-s − 0.246·10-s − 1.00·11-s − 1.24·12-s − 1.52·13-s − 0.799·15-s + 0.179·16-s − 0.617·17-s + 1.21·18-s − 0.631·19-s + 0.311·20-s − 0.557·22-s + 0.482·23-s − 1.67·24-s + 0.200·25-s − 0.843·26-s + 2.13·27-s − 1.23·29-s − 0.441·30-s + 0.588·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.12T + 32T^{2} \) |
| 3 | \( 1 - 27.8T + 243T^{2} \) |
| 11 | \( 1 + 405.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 931.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 736.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 994.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.61e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.46e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.45e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.76e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.88e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.90e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.49e5T + 8.58e9T^{2} \) |
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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
−10.47718773597526320229033447620, −9.529866627903875164980656116759, −8.754231047178464838395556422961, −7.947659446049147532913330313179, −7.06322217680693363120351184140, −5.12510116462414021518067263054, −4.22747393210658824808671607627, −3.16387213929912410247230503341, −2.23991617151723724419948030948, 0,
2.23991617151723724419948030948, 3.16387213929912410247230503341, 4.22747393210658824808671607627, 5.12510116462414021518067263054, 7.06322217680693363120351184140, 7.947659446049147532913330313179, 8.754231047178464838395556422961, 9.529866627903875164980656116759, 10.47718773597526320229033447620
