| L(s) = 1 | − 32.0·2-s + 511.·3-s − 7.16e3·4-s − 6.26e4·5-s − 1.63e4·6-s − 5.16e5·7-s + 4.91e5·8-s − 1.33e6·9-s + 2.00e6·10-s + 2.85e6·11-s − 3.66e6·12-s + 1.15e7·13-s + 1.65e7·14-s − 3.20e7·15-s + 4.29e7·16-s − 8.05e7·17-s + 4.26e7·18-s − 3.82e8·19-s + 4.48e8·20-s − 2.64e8·21-s − 9.13e7·22-s + 1.48e8·23-s + 2.51e8·24-s + 2.70e9·25-s − 3.68e8·26-s − 1.49e9·27-s + 3.70e9·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.405·3-s − 0.874·4-s − 1.79·5-s − 0.143·6-s − 1.65·7-s + 0.663·8-s − 0.835·9-s + 0.634·10-s + 0.485·11-s − 0.354·12-s + 0.660·13-s + 0.587·14-s − 0.726·15-s + 0.640·16-s − 0.809·17-s + 0.295·18-s − 1.86·19-s + 1.56·20-s − 0.672·21-s − 0.171·22-s + 0.208·23-s + 0.268·24-s + 2.21·25-s − 0.233·26-s − 0.743·27-s + 1.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.2133729427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2133729427\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 1.48e8T \) |
| good | 2 | \( 1 + 32.0T + 8.19e3T^{2} \) |
| 3 | \( 1 - 511.T + 1.59e6T^{2} \) |
| 5 | \( 1 + 6.26e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.16e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 2.85e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.15e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 8.05e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.82e8T + 4.20e16T^{2} \) |
| 29 | \( 1 - 3.69e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 1.91e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 9.74e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.00e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 3.06e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 5.09e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.36e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 4.59e9T + 1.04e23T^{2} \) |
| 61 | \( 1 + 2.49e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 4.16e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.95e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.27e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.18e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.35e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 7.69e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 9.67e12T + 6.73e25T^{2} \) |
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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
−14.92003959967837089790133230336, −13.40083832450599953031866818014, −12.28795678904415973209725351602, −10.73802837166310672855388183356, −8.995461899905894933738530791645, −8.329854490617703862336181177909, −6.62204779337689799370159256291, −4.20271955999581217879643864031, −3.27814884252729412972817593452, −0.29980454613855238582035024629,
0.29980454613855238582035024629, 3.27814884252729412972817593452, 4.20271955999581217879643864031, 6.62204779337689799370159256291, 8.329854490617703862336181177909, 8.995461899905894933738530791645, 10.73802837166310672855388183356, 12.28795678904415973209725351602, 13.40083832450599953031866818014, 14.92003959967837089790133230336
