| L(s) = 1 | − 0.618·2-s + 1.23·3-s − 1.61·4-s + 5-s − 0.763·6-s − 4.23·7-s + 2.23·8-s − 1.47·9-s − 0.618·10-s + 2·11-s − 2.00·12-s + 1.23·13-s + 2.61·14-s + 1.23·15-s + 1.85·16-s + 5.23·17-s + 0.909·18-s + 2.23·19-s − 1.61·20-s − 5.23·21-s − 1.23·22-s − 7.70·23-s + 2.76·24-s − 4·25-s − 0.763·26-s − 5.52·27-s + 6.85·28-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.713·3-s − 0.809·4-s + 0.447·5-s − 0.311·6-s − 1.60·7-s + 0.790·8-s − 0.490·9-s − 0.195·10-s + 0.603·11-s − 0.577·12-s + 0.342·13-s + 0.699·14-s + 0.319·15-s + 0.463·16-s + 1.26·17-s + 0.214·18-s + 0.512·19-s − 0.361·20-s − 1.14·21-s − 0.263·22-s − 1.60·23-s + 0.564·24-s − 0.800·25-s − 0.149·26-s − 1.06·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6021252350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6021252350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 1.94T + 97T^{2} \) |
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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
−17.03606772532803878991628196034, −15.99097596480158213236404952174, −14.20506575973747679651369846082, −13.64320749624703022846928675733, −12.27803936712599513177270754436, −9.988884574157259088769712815632, −9.392764051520264102402757332269, −8.066873544053878311165268740079, −6.03278399652487520131783270143, −3.50551568171874308730107016287,
3.50551568171874308730107016287, 6.03278399652487520131783270143, 8.066873544053878311165268740079, 9.392764051520264102402757332269, 9.988884574157259088769712815632, 12.27803936712599513177270754436, 13.64320749624703022846928675733, 14.20506575973747679651369846082, 15.99097596480158213236404952174, 17.03606772532803878991628196034
