| L(s) = 1 | − 0.319·5-s + 2.61·7-s + 4.31·11-s + 6.51·13-s + 4.19·17-s + 19-s − 0.639·23-s − 4.89·25-s − 7.87·29-s + 6·31-s − 0.837·35-s + 4·37-s − 12.5·41-s + 1.38·43-s + 7.55·47-s − 0.137·49-s − 13.1·53-s − 1.38·55-s + 13.0·59-s − 5.89·61-s − 2.08·65-s − 11.7·67-s − 11.7·71-s + 15.1·73-s + 11.3·77-s − 0.517·79-s + 11.8·83-s + ⋯ |
| L(s) = 1 | − 0.142·5-s + 0.990·7-s + 1.30·11-s + 1.80·13-s + 1.01·17-s + 0.229·19-s − 0.133·23-s − 0.979·25-s − 1.46·29-s + 1.07·31-s − 0.141·35-s + 0.657·37-s − 1.95·41-s + 0.210·43-s + 1.10·47-s − 0.0196·49-s − 1.80·53-s − 0.186·55-s + 1.69·59-s − 0.755·61-s − 0.258·65-s − 1.43·67-s − 1.39·71-s + 1.77·73-s + 1.28·77-s − 0.0582·79-s + 1.30·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488909586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488909586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.319T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 - 6.51T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 23 | \( 1 + 0.639T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 0.517T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.761046875486362447512216349024, −8.080388142679842340405362853422, −7.48346510968021527659219490038, −6.36662955804707588671859888085, −5.88552442360753240033619582943, −4.88451012755515859520362620232, −3.91038754575440222447577772839, −3.42423803188520110977604464463, −1.80346018474138264058949249081, −1.12086061654203385707645960486,
1.12086061654203385707645960486, 1.80346018474138264058949249081, 3.42423803188520110977604464463, 3.91038754575440222447577772839, 4.88451012755515859520362620232, 5.88552442360753240033619582943, 6.36662955804707588671859888085, 7.48346510968021527659219490038, 8.080388142679842340405362853422, 8.761046875486362447512216349024
