| L(s) = 1 | + 0.0881·2-s − 3.20·3-s − 1.99·4-s + 5-s − 0.282·6-s − 1.95·7-s − 0.351·8-s + 7.26·9-s + 0.0881·10-s − 11-s + 6.38·12-s + 3.20·13-s − 0.172·14-s − 3.20·15-s + 3.95·16-s + 0.503·17-s + 0.640·18-s + 19-s − 1.99·20-s + 6.25·21-s − 0.0881·22-s + 1.05·23-s + 1.12·24-s + 25-s + 0.282·26-s − 13.6·27-s + 3.88·28-s + ⋯ |
| L(s) = 1 | + 0.0623·2-s − 1.84·3-s − 0.996·4-s + 0.447·5-s − 0.115·6-s − 0.737·7-s − 0.124·8-s + 2.42·9-s + 0.0278·10-s − 0.301·11-s + 1.84·12-s + 0.889·13-s − 0.0459·14-s − 0.827·15-s + 0.988·16-s + 0.122·17-s + 0.150·18-s + 0.229·19-s − 0.445·20-s + 1.36·21-s − 0.0187·22-s + 0.220·23-s + 0.230·24-s + 0.200·25-s + 0.0554·26-s − 2.62·27-s + 0.734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0881T + 2T^{2} \) |
| 3 | \( 1 + 3.20T + 3T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 13 | \( 1 - 3.20T + 13T^{2} \) |
| 17 | \( 1 - 0.503T + 17T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 - 9.52T + 31T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 + 9.39T + 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 + 1.53T + 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 + 1.55T + 73T^{2} \) |
| 79 | \( 1 - 3.00T + 79T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 - 6.10T + 89T^{2} \) |
| 97 | \( 1 + 4.69T + 97T^{2} \) |
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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
−9.808584307325280268850538388518, −8.903310233301945018355952283008, −7.70747774896139184906052044713, −6.57031995808677251812175765834, −5.99488277320995693797376354037, −5.27799974047905390366857456996, −4.50551575743991790104397101702, −3.42197831029460537931350324079, −1.28254953411861682805164698558, 0,
1.28254953411861682805164698558, 3.42197831029460537931350324079, 4.50551575743991790104397101702, 5.27799974047905390366857456996, 5.99488277320995693797376354037, 6.57031995808677251812175765834, 7.70747774896139184906052044713, 8.903310233301945018355952283008, 9.808584307325280268850538388518
