| L(s) = 1 | − 66.7·2-s − 326.·3-s + 2.40e3·4-s + 1.09e4·5-s + 2.17e4·6-s − 7.43e4·7-s − 2.37e4·8-s − 7.05e4·9-s − 7.30e5·10-s − 2.38e5·11-s − 7.84e5·12-s − 7.07e4·13-s + 4.96e6·14-s − 3.57e6·15-s − 3.33e6·16-s + 4.61e6·17-s + 4.71e6·18-s − 1.27e7·19-s + 2.63e7·20-s + 2.42e7·21-s + 1.59e7·22-s − 6.43e6·23-s + 7.75e6·24-s + 7.11e7·25-s + 4.72e6·26-s + 8.08e7·27-s − 1.78e8·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 0.775·3-s + 1.17·4-s + 1.56·5-s + 1.14·6-s − 1.67·7-s − 0.256·8-s − 0.398·9-s − 2.31·10-s − 0.447·11-s − 0.910·12-s − 0.0528·13-s + 2.46·14-s − 1.21·15-s − 0.795·16-s + 0.787·17-s + 0.587·18-s − 1.17·19-s + 1.84·20-s + 1.29·21-s + 0.659·22-s − 0.208·23-s + 0.198·24-s + 1.45·25-s + 0.0779·26-s + 1.08·27-s − 1.96·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.5089624512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5089624512\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 6.43e6T \) |
| good | 2 | \( 1 + 66.7T + 2.04e3T^{2} \) |
| 3 | \( 1 + 326.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 1.09e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 7.43e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 2.38e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 7.07e4T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.61e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.27e7T + 1.16e14T^{2} \) |
| 29 | \( 1 - 6.04e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 9.27e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.79e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 4.76e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 9.81e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.43e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.96e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 6.00e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.58e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.75e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.77e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.55e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.78e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.84e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.32e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.30e10T + 7.15e21T^{2} \) |
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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
−15.99562242241962940066257808982, −13.80117547686228037072268696485, −12.51699064822073357989742597886, −10.54479843923238630960111679682, −9.952901853223047149586331305125, −8.826187173664338485617395045487, −6.71685229553181147299528203972, −5.74800795153184125578138072190, −2.45946709252111215422519490880, −0.63039921227013152568507738084,
0.63039921227013152568507738084, 2.45946709252111215422519490880, 5.74800795153184125578138072190, 6.71685229553181147299528203972, 8.826187173664338485617395045487, 9.952901853223047149586331305125, 10.54479843923238630960111679682, 12.51699064822073357989742597886, 13.80117547686228037072268696485, 15.99562242241962940066257808982
