| L(s) = 1 | + 2.39e3·2-s + 3.64e6·4-s − 9.76e6·5-s + 8.95e8·7-s + 3.69e9·8-s − 2.33e10·10-s + 9.77e10·11-s − 8.68e11·13-s + 2.14e12·14-s + 1.22e12·16-s − 5.17e11·17-s + 4.21e13·19-s − 3.55e13·20-s + 2.34e14·22-s + 3.07e14·23-s + 9.53e13·25-s − 2.08e15·26-s + 3.25e15·28-s + 1.24e15·29-s − 9.49e14·31-s − 4.82e15·32-s − 1.24e15·34-s − 8.74e15·35-s + 4.84e15·37-s + 1.00e17·38-s − 3.61e16·40-s + 1.03e17·41-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.73·4-s − 0.447·5-s + 1.19·7-s + 1.21·8-s − 0.739·10-s + 1.13·11-s − 1.74·13-s + 1.98·14-s + 0.278·16-s − 0.0622·17-s + 1.57·19-s − 0.776·20-s + 1.88·22-s + 1.54·23-s + 0.199·25-s − 2.89·26-s + 2.07·28-s + 0.548·29-s − 0.208·31-s − 0.757·32-s − 0.103·34-s − 0.535·35-s + 0.165·37-s + 2.60·38-s − 0.544·40-s + 1.20·41-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
| \(L(11)\) |
\(\approx\) |
\(7.302634282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.302634282\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 2.39e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 8.95e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 9.77e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 8.68e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 5.17e11T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.21e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.07e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.24e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 9.49e14T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.84e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.03e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 6.02e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.18e16T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.86e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 7.16e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 3.96e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.14e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.31e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.81e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.32e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.04e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.36e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 4.85e20T + 5.27e41T^{2} \) |
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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
−11.80474859675316213786298013152, −11.18435898877125472794369655865, −9.367521428896148039255941039408, −7.67713575976551129971450841988, −6.77580345857464314210887521041, −5.17745802409909081054781858951, −4.70747926316199739286147433175, −3.49878089601175179437164702551, −2.36458958221560632926128730066, −1.02713719181479015870592617451,
1.02713719181479015870592617451, 2.36458958221560632926128730066, 3.49878089601175179437164702551, 4.70747926316199739286147433175, 5.17745802409909081054781858951, 6.77580345857464314210887521041, 7.67713575976551129971450841988, 9.367521428896148039255941039408, 11.18435898877125472794369655865, 11.80474859675316213786298013152
