| L(s) = 1 | − 1.15e3·2-s − 5.90e4·3-s − 7.63e5·4-s + 6.81e7·6-s − 5.42e7·7-s + 3.30e9·8-s + 3.48e9·9-s − 9.88e10·11-s + 4.50e10·12-s − 3.35e10·13-s + 6.26e10·14-s − 2.21e12·16-s − 1.60e13·17-s − 4.02e12·18-s − 1.23e13·19-s + 3.20e12·21-s + 1.14e14·22-s + 3.49e14·23-s − 1.95e14·24-s + 3.87e13·26-s − 2.05e14·27-s + 4.13e13·28-s + 9.02e14·29-s − 3.33e15·31-s − 4.37e15·32-s + 5.83e15·33-s + 1.84e16·34-s + ⋯ |
| L(s) = 1 | − 0.797·2-s − 0.577·3-s − 0.364·4-s + 0.460·6-s − 0.0725·7-s + 1.08·8-s + 0.333·9-s − 1.14·11-s + 0.210·12-s − 0.0674·13-s + 0.0578·14-s − 0.503·16-s − 1.92·17-s − 0.265·18-s − 0.461·19-s + 0.0418·21-s + 0.916·22-s + 1.75·23-s − 0.628·24-s + 0.0538·26-s − 0.192·27-s + 0.0264·28-s + 0.398·29-s − 0.730·31-s − 0.686·32-s + 0.663·33-s + 1.53·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.2291046713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2291046713\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.15e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 5.42e7T + 5.58e17T^{2} \) |
| 11 | \( 1 + 9.88e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.35e10T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.60e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.23e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.49e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 9.02e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 3.33e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.76e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.62e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.86e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.66e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.04e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.49e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.01e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.39e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 1.39e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 3.21e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 5.59e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.95e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 4.40e18T + 8.65e40T^{2} \) |
| 97 | \( 1 - 6.20e20T + 5.27e41T^{2} \) |
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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
−10.67358205155966805977716712659, −9.524685434690945634302992578357, −8.633963199556947080711244993031, −7.56315966065699932356393116734, −6.49740652602646053611846949525, −5.07128622371812045854404234371, −4.38927678989063332310916925135, −2.70333638464915024684553673196, −1.44691291187077919619540540673, −0.24274216492944101092101369077,
0.24274216492944101092101369077, 1.44691291187077919619540540673, 2.70333638464915024684553673196, 4.38927678989063332310916925135, 5.07128622371812045854404234371, 6.49740652602646053611846949525, 7.56315966065699932356393116734, 8.633963199556947080711244993031, 9.524685434690945634302992578357, 10.67358205155966805977716712659
