| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 5.23·7-s + 8-s + 9-s − 3.85·11-s + 12-s + 0.381·13-s + 5.23·14-s + 16-s − 5.61·17-s + 18-s + 4.47·19-s + 5.23·21-s − 3.85·22-s − 4.61·23-s + 24-s + 0.381·26-s + 27-s + 5.23·28-s + 3.61·29-s − 8.32·31-s + 32-s − 3.85·33-s − 5.61·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.97·7-s + 0.353·8-s + 0.333·9-s − 1.16·11-s + 0.288·12-s + 0.105·13-s + 1.39·14-s + 0.250·16-s − 1.36·17-s + 0.235·18-s + 1.02·19-s + 1.14·21-s − 0.821·22-s − 0.962·23-s + 0.204·24-s + 0.0749·26-s + 0.192·27-s + 0.989·28-s + 0.671·29-s − 1.49·31-s + 0.176·32-s − 0.670·33-s − 0.963·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.194951242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.194951242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 0.381T + 13T^{2} \) |
| 17 | \( 1 + 5.61T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 4.61T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 8.32T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 1.52T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 + 9.23T + 97T^{2} \) |
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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
−10.68726466988108192061285123791, −9.473669788470333417785331177151, −8.284529412904579796978929558959, −7.918568401952119951221905921904, −6.99747737981336338105123454161, −5.58109780506102895101606585791, −4.87573687503860074393076887450, −4.05824444479862422632633383052, −2.62088114866603900948906255808, −1.71240608245733582826171948694,
1.71240608245733582826171948694, 2.62088114866603900948906255808, 4.05824444479862422632633383052, 4.87573687503860074393076887450, 5.58109780506102895101606585791, 6.99747737981336338105123454161, 7.918568401952119951221905921904, 8.284529412904579796978929558959, 9.473669788470333417785331177151, 10.68726466988108192061285123791
