| L(s) = 1 | − 4.56·2-s − 1.63·3-s − 11.1·4-s − 103.·5-s + 7.43·6-s + 126.·7-s + 197.·8-s − 240.·9-s + 471.·10-s − 14.8·11-s + 18.2·12-s − 169·13-s − 576.·14-s + 168.·15-s − 540.·16-s − 1.05e3·17-s + 1.09e3·18-s − 213.·19-s + 1.15e3·20-s − 205.·21-s + 67.6·22-s − 4.23e3·23-s − 321.·24-s + 7.57e3·25-s + 770.·26-s + 788.·27-s − 1.41e3·28-s + ⋯ |
| L(s) = 1 | − 0.806·2-s − 0.104·3-s − 0.349·4-s − 1.85·5-s + 0.0843·6-s + 0.974·7-s + 1.08·8-s − 0.989·9-s + 1.49·10-s − 0.0369·11-s + 0.0365·12-s − 0.277·13-s − 0.785·14-s + 0.193·15-s − 0.527·16-s − 0.882·17-s + 0.797·18-s − 0.135·19-s + 0.647·20-s − 0.101·21-s + 0.0297·22-s − 1.66·23-s − 0.113·24-s + 2.42·25-s + 0.223·26-s + 0.208·27-s − 0.340·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + 169T \) |
| good | 2 | \( 1 + 4.56T + 32T^{2} \) |
| 3 | \( 1 + 1.63T + 243T^{2} \) |
| 5 | \( 1 + 103.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 126.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 14.8T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 213.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 504.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.51e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.80e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.81e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.05e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.79e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−18.21641402743395835008708639836, −17.04546383003913073150984208649, −15.62452668382559574219879508070, −14.19373837112560839413522349369, −11.96857811561027895312429109720, −10.89265740864086131542400088784, −8.604094016517240757911341303022, −7.76406369095647844737940227415, −4.42812287454429366295353399292, 0,
4.42812287454429366295353399292, 7.76406369095647844737940227415, 8.604094016517240757911341303022, 10.89265740864086131542400088784, 11.96857811561027895312429109720, 14.19373837112560839413522349369, 15.62452668382559574219879508070, 17.04546383003913073150984208649, 18.21641402743395835008708639836
