L(s) = 1 | + 1.54·2-s + 0.456·3-s + 0.402·4-s + 0.324·5-s + 0.707·6-s + 2.73·7-s − 2.47·8-s − 2.79·9-s + 0.502·10-s + 5.58·11-s + 0.183·12-s + 6.14·13-s + 4.24·14-s + 0.147·15-s − 4.64·16-s − 1.80·17-s − 4.32·18-s + 1.89·19-s + 0.130·20-s + 1.24·21-s + 8.65·22-s + 0.651·23-s − 1.13·24-s − 4.89·25-s + 9.53·26-s − 2.64·27-s + 1.10·28-s + ⋯ |
L(s) = 1 | + 1.09·2-s + 0.263·3-s + 0.201·4-s + 0.144·5-s + 0.288·6-s + 1.03·7-s − 0.875·8-s − 0.930·9-s + 0.158·10-s + 1.68·11-s + 0.0530·12-s + 1.70·13-s + 1.13·14-s + 0.0382·15-s − 1.16·16-s − 0.437·17-s − 1.01·18-s + 0.435·19-s + 0.0291·20-s + 0.272·21-s + 1.84·22-s + 0.135·23-s − 0.230·24-s − 0.978·25-s + 1.86·26-s − 0.508·27-s + 0.208·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.894678058\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.894678058\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 619 | \( 1 - T \) |
good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 - 0.456T + 3T^{2} \) |
| 5 | \( 1 - 0.324T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 - 6.14T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 - 1.89T + 19T^{2} \) |
| 23 | \( 1 - 0.651T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 - 7.40T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 8.46T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 + 8.37T + 53T^{2} \) |
| 59 | \( 1 - 8.13T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 0.0636T + 67T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 0.958T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 4.18T + 97T^{2} \) |
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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
−11.09798487317286711209584345696, −9.598520926101837601765599195107, −8.667180458813392802991198341913, −8.304921836189942097797184912026, −6.60820308590946192113980955395, −6.01571767052915364928022900975, −4.97908343385799248747526347199, −4.00365536488285460152890386671, −3.21450652158866582355242683996, −1.56667067300482226653643235754,
1.56667067300482226653643235754, 3.21450652158866582355242683996, 4.00365536488285460152890386671, 4.97908343385799248747526347199, 6.01571767052915364928022900975, 6.60820308590946192113980955395, 8.304921836189942097797184912026, 8.667180458813392802991198341913, 9.598520926101837601765599195107, 11.09798487317286711209584345696