| L(s) = 1 | + 1.21·3-s + 2.91·5-s + 2.62·7-s − 1.51·9-s − 0.848·11-s + 3.08·13-s + 3.55·15-s + 2.86·17-s + 1.56·19-s + 3.19·21-s − 3.64·23-s + 3.51·25-s − 5.50·27-s − 5.07·29-s + 7.22·31-s − 1.03·33-s + 7.64·35-s + 3.70·37-s + 3.76·39-s − 12.6·41-s + 4.68·43-s − 4.41·45-s + 13.0·47-s − 0.124·49-s + 3.49·51-s + 11.3·53-s − 2.47·55-s + ⋯ |
| L(s) = 1 | + 0.704·3-s + 1.30·5-s + 0.991·7-s − 0.504·9-s − 0.255·11-s + 0.856·13-s + 0.918·15-s + 0.695·17-s + 0.358·19-s + 0.697·21-s − 0.759·23-s + 0.702·25-s − 1.05·27-s − 0.942·29-s + 1.29·31-s − 0.180·33-s + 1.29·35-s + 0.609·37-s + 0.603·39-s − 1.97·41-s + 0.714·43-s − 0.657·45-s + 1.89·47-s − 0.0178·49-s + 0.489·51-s + 1.56·53-s − 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.123798103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.123798103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 1.21T + 3T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 + 0.848T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 - 2.86T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 - 7.22T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 1.66T + 59T^{2} \) |
| 61 | \( 1 - 5.08T + 61T^{2} \) |
| 67 | \( 1 + 9.12T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.115657168217859092664775860262, −7.25509006801928315838173024888, −6.30771830522055561888336094071, −5.60901553347981184876810683186, −5.32912733466053547015818120255, −4.21165976461627999350877728247, −3.41385255586668536145524928780, −2.48028967249677577037878360698, −1.94074463972590162753340516718, −1.02478772336947078585287280405,
1.02478772336947078585287280405, 1.94074463972590162753340516718, 2.48028967249677577037878360698, 3.41385255586668536145524928780, 4.21165976461627999350877728247, 5.32912733466053547015818120255, 5.60901553347981184876810683186, 6.30771830522055561888336094071, 7.25509006801928315838173024888, 8.115657168217859092664775860262
