| L(s) = 1 | − 61.2·2-s − 243·3-s + 1.70e3·4-s + 1.48e4·6-s − 7.69e4·7-s + 2.09e4·8-s + 5.90e4·9-s + 1.90e5·11-s − 4.14e5·12-s − 6.79e4·13-s + 4.71e6·14-s − 4.77e6·16-s − 8.85e6·17-s − 3.61e6·18-s − 2.90e6·19-s + 1.87e7·21-s − 1.16e7·22-s + 4.16e7·23-s − 5.09e6·24-s + 4.16e6·26-s − 1.43e7·27-s − 1.31e8·28-s + 1.13e8·29-s + 2.80e8·31-s + 2.49e8·32-s − 4.62e7·33-s + 5.42e8·34-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 0.577·3-s + 0.833·4-s + 0.781·6-s − 1.73·7-s + 0.226·8-s + 0.333·9-s + 0.356·11-s − 0.480·12-s − 0.0507·13-s + 2.34·14-s − 1.13·16-s − 1.51·17-s − 0.451·18-s − 0.268·19-s + 0.999·21-s − 0.482·22-s + 1.35·23-s − 0.130·24-s + 0.0687·26-s − 0.192·27-s − 1.44·28-s + 1.02·29-s + 1.76·31-s + 1.31·32-s − 0.205·33-s + 2.04·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 61.2T + 2.04e3T^{2} \) |
| 7 | \( 1 + 7.69e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.90e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 6.79e4T + 1.79e12T^{2} \) |
| 17 | \( 1 + 8.85e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 2.90e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.16e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.13e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.80e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.37e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.45e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 4.54e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 4.50e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.94e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 4.37e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 7.29e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.93e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.40e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 7.94e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.13e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.51e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.91e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 2.39e10T + 7.15e21T^{2} \) |
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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
−11.35036316955076537869736976582, −10.31691864926062326520247272808, −9.499147539062087796281494400186, −8.608052809895160408195083960308, −6.96473759483848226963882718218, −6.38199568969072517017899566165, −4.44985913701821078349056507940, −2.68891021611310021249495678799, −0.944175523249201729112713479307, 0,
0.944175523249201729112713479307, 2.68891021611310021249495678799, 4.44985913701821078349056507940, 6.38199568969072517017899566165, 6.96473759483848226963882718218, 8.608052809895160408195083960308, 9.499147539062087796281494400186, 10.31691864926062326520247272808, 11.35036316955076537869736976582
