L(s) = 1 | + 60.1·2-s − 243·3-s + 1.57e3·4-s − 9.25e3·5-s − 1.46e4·6-s − 7.12e3·7-s − 2.85e4·8-s + 5.90e4·9-s − 5.57e5·10-s + 2.04e4·11-s − 3.82e5·12-s + 2.48e6·13-s − 4.28e5·14-s + 2.24e6·15-s − 4.94e6·16-s + 5.23e6·17-s + 3.55e6·18-s + 6.43e6·19-s − 1.45e7·20-s + 1.73e6·21-s + 1.22e6·22-s + 2.55e7·23-s + 6.93e6·24-s + 3.68e7·25-s + 1.49e8·26-s − 1.43e7·27-s − 1.12e7·28-s + ⋯ |
L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.768·4-s − 1.32·5-s − 0.767·6-s − 0.160·7-s − 0.307·8-s + 0.333·9-s − 1.76·10-s + 0.0382·11-s − 0.443·12-s + 1.85·13-s − 0.212·14-s + 0.764·15-s − 1.17·16-s + 0.893·17-s + 0.443·18-s + 0.595·19-s − 1.01·20-s + 0.0924·21-s + 0.0508·22-s + 0.829·23-s + 0.177·24-s + 0.755·25-s + 2.46·26-s − 0.192·27-s − 0.123·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 243T \) |
| 59 | \( 1 - 7.14e8T \) |
good | 2 | \( 1 - 60.1T + 2.04e3T^{2} \) |
| 5 | \( 1 + 9.25e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 7.12e3T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.04e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.48e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.23e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.43e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.55e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.12e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.37e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.05e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.64e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 4.14e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.42e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 6.03e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 4.98e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.57e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.25e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 9.52e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.72e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.80e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.01e11T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.67e10T + 7.15e21T^{2} \) |
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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
−10.69381131306020577066031331762, −9.074884900658281219866393837263, −7.921575327668569885791646851599, −6.75578421089551345383665107286, −5.79358765807130889365315654356, −4.81967609883974333071895720913, −3.71715707824079507198615163864, −3.29014416365077368936844298644, −1.22944224924534102777109311462, 0,
1.22944224924534102777109311462, 3.29014416365077368936844298644, 3.71715707824079507198615163864, 4.81967609883974333071895720913, 5.79358765807130889365315654356, 6.75578421089551345383665107286, 7.921575327668569885791646851599, 9.074884900658281219866393837263, 10.69381131306020577066031331762