| L(s) = 1 | + 2.50·2-s + 2.36·3-s + 4.25·4-s + 5-s + 5.92·6-s − 2.67·7-s + 5.64·8-s + 2.60·9-s + 2.50·10-s + 11-s + 10.0·12-s + 3.48·13-s − 6.68·14-s + 2.36·15-s + 5.60·16-s + 4.44·17-s + 6.52·18-s − 4.04·19-s + 4.25·20-s − 6.32·21-s + 2.50·22-s + 5.55·23-s + 13.3·24-s + 25-s + 8.71·26-s − 0.930·27-s − 11.3·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 1.36·3-s + 2.12·4-s + 0.447·5-s + 2.41·6-s − 1.00·7-s + 1.99·8-s + 0.868·9-s + 0.790·10-s + 0.301·11-s + 2.90·12-s + 0.966·13-s − 1.78·14-s + 0.611·15-s + 1.40·16-s + 1.07·17-s + 1.53·18-s − 0.927·19-s + 0.951·20-s − 1.38·21-s + 0.533·22-s + 1.15·23-s + 2.72·24-s + 0.200·25-s + 1.70·26-s − 0.179·27-s − 2.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.543087910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.543087910\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 3 | \( 1 - 2.36T + 3T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 - 0.596T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 - 6.60T + 43T^{2} \) |
| 47 | \( 1 - 0.172T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 + 4.38T + 59T^{2} \) |
| 61 | \( 1 - 2.23T + 61T^{2} \) |
| 67 | \( 1 - 5.08T + 67T^{2} \) |
| 71 | \( 1 + 3.91T + 71T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 8.54T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 8.71T + 97T^{2} \) |
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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
−8.408004683739896929252153176512, −7.45890290881075095212017274478, −6.83013930719230012623354908234, −6.00243830223916062161775817956, −5.58101823721906115251839085500, −4.37748368780119578474178109110, −3.67310162036144320995197999909, −3.19242056566737477143776056595, −2.51769155085796141277037979361, −1.53450754409896208109093325712,
1.53450754409896208109093325712, 2.51769155085796141277037979361, 3.19242056566737477143776056595, 3.67310162036144320995197999909, 4.37748368780119578474178109110, 5.58101823721906115251839085500, 6.00243830223916062161775817956, 6.83013930719230012623354908234, 7.45890290881075095212017274478, 8.408004683739896929252153176512
