| L(s) = 1 | − 2·5-s + 1.53·7-s − 6.05·11-s − 5.50·13-s − 1.11·17-s − 2.87·19-s − 6.59·23-s − 25-s − 3.97·29-s − 1.23·31-s − 3.07·35-s + 11.1·37-s − 2.55·41-s + 11.0·43-s − 8.14·47-s − 4.63·49-s + 6.62·53-s + 12.1·55-s + 12.3·59-s − 0.918·61-s + 11.0·65-s + 12.9·67-s + 11.3·71-s + 5.47·73-s − 9.30·77-s + 10.1·79-s − 14.5·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.580·7-s − 1.82·11-s − 1.52·13-s − 0.270·17-s − 0.658·19-s − 1.37·23-s − 0.200·25-s − 0.737·29-s − 0.221·31-s − 0.519·35-s + 1.83·37-s − 0.398·41-s + 1.68·43-s − 1.18·47-s − 0.662·49-s + 0.910·53-s + 1.63·55-s + 1.60·59-s − 0.117·61-s + 1.36·65-s + 1.58·67-s + 1.35·71-s + 0.641·73-s − 1.05·77-s + 1.14·79-s − 1.59·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6188541459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6188541459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 6.05T + 11T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 + 3.97T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 - 6.62T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.918T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 5.47T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 4.26T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−7.960198100914725017308485325657, −7.63446478764609642980209596232, −6.86665813890967937709090495849, −5.77087686705462933241217573730, −5.15204456274675223492729330345, −4.45503180300309749304099542893, −3.79730173275278646819600034149, −2.53887814660390827700433733198, −2.17174007771652196462064891864, −0.38301276128921276321183727139,
0.38301276128921276321183727139, 2.17174007771652196462064891864, 2.53887814660390827700433733198, 3.79730173275278646819600034149, 4.45503180300309749304099542893, 5.15204456274675223492729330345, 5.77087686705462933241217573730, 6.86665813890967937709090495849, 7.63446478764609642980209596232, 7.960198100914725017308485325657
