| L(s) = 1 | − 9.99e2·2-s − 1.09e6·4-s − 9.76e6·5-s + 9.82e7·7-s + 3.19e9·8-s + 9.76e9·10-s + 9.60e10·11-s + 6.22e11·13-s − 9.82e10·14-s − 8.91e11·16-s + 1.42e13·17-s − 8.28e12·19-s + 1.07e13·20-s − 9.60e13·22-s + 1.17e14·23-s + 9.53e13·25-s − 6.21e14·26-s − 1.07e14·28-s + 1.98e15·29-s − 5.31e15·31-s − 5.80e15·32-s − 1.42e16·34-s − 9.59e14·35-s + 1.99e16·37-s + 8.28e15·38-s − 3.11e16·40-s + 2.99e16·41-s + ⋯ |
| L(s) = 1 | − 0.690·2-s − 0.523·4-s − 0.447·5-s + 0.131·7-s + 1.05·8-s + 0.308·10-s + 1.11·11-s + 1.25·13-s − 0.0907·14-s − 0.202·16-s + 1.71·17-s − 0.310·19-s + 0.234·20-s − 0.771·22-s + 0.591·23-s + 0.199·25-s − 0.864·26-s − 0.0687·28-s + 0.877·29-s − 1.16·31-s − 0.911·32-s − 1.18·34-s − 0.0587·35-s + 0.681·37-s + 0.214·38-s − 0.470·40-s + 0.348·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.602221009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.602221009\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 9.76e6T \) |
| good | 2 | \( 1 + 9.99e2T + 2.09e6T^{2} \) |
| 7 | \( 1 - 9.82e7T + 5.58e17T^{2} \) |
| 11 | \( 1 - 9.60e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 6.22e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.42e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 8.28e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.17e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.98e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 5.31e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.99e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 2.99e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 3.44e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.23e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.54e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 6.81e17T + 1.54e37T^{2} \) |
| 61 | \( 1 + 3.28e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.07e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.71e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 6.79e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.28e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.24e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.94e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 4.69e20T + 5.27e41T^{2} \) |
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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
−11.43914265969001930955053230393, −10.33345155017692909786317624407, −9.157216901054178425222219977391, −8.355381688447850844467796732603, −7.25804940060755980960109332552, −5.78012521229111048927917679296, −4.32942967644414584371168425737, −3.41340509201448685703420026433, −1.41172469132609629288555915333, −0.75906779354436926969320585849,
0.75906779354436926969320585849, 1.41172469132609629288555915333, 3.41340509201448685703420026433, 4.32942967644414584371168425737, 5.78012521229111048927917679296, 7.25804940060755980960109332552, 8.355381688447850844467796732603, 9.157216901054178425222219977391, 10.33345155017692909786317624407, 11.43914265969001930955053230393
