L(s) = 1 | − 2·2-s + 6.12·3-s + 4·4-s + 18.7·5-s − 12.2·6-s − 21.9·7-s − 8·8-s + 10.5·9-s − 37.4·10-s + 57.7·11-s + 24.5·12-s − 18.6·13-s + 43.8·14-s + 114.·15-s + 16·16-s − 44.5·17-s − 21.1·18-s + 128.·19-s + 74.9·20-s − 134.·21-s − 115.·22-s − 112.·23-s − 49.0·24-s + 226.·25-s + 37.2·26-s − 100.·27-s − 87.6·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.17·3-s + 0.5·4-s + 1.67·5-s − 0.833·6-s − 1.18·7-s − 0.353·8-s + 0.390·9-s − 1.18·10-s + 1.58·11-s + 0.589·12-s − 0.397·13-s + 0.836·14-s + 1.97·15-s + 0.250·16-s − 0.635·17-s − 0.276·18-s + 1.55·19-s + 0.838·20-s − 1.39·21-s − 1.11·22-s − 1.01·23-s − 0.416·24-s + 1.81·25-s + 0.281·26-s − 0.718·27-s − 0.591·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.786850710\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.786850710\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 37 | \( 1 + 37T \) |
good | 3 | \( 1 - 6.12T + 27T^{2} \) |
| 5 | \( 1 - 18.7T + 125T^{2} \) |
| 7 | \( 1 + 21.9T + 343T^{2} \) |
| 11 | \( 1 - 57.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 127.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.8T + 2.97e4T^{2} \) |
| 41 | \( 1 + 438.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 0.358T + 1.48e5T^{2} \) |
| 59 | \( 1 - 698.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 494.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 525.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 14.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 980.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 273.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 559.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 77.1T + 9.12e5T^{2} \) |
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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
−13.95618726829362024059731865348, −13.39178489842652739710856462010, −11.90054603684560371513123506662, −9.991396882815858988684437517264, −9.502657276706303082172623972497, −8.806655653872362639529637943363, −7.02332745766936583040110129147, −5.96372402165584654416789887286, −3.27368114020858683240359520402, −1.86114520505421590067682439614,
1.86114520505421590067682439614, 3.27368114020858683240359520402, 5.96372402165584654416789887286, 7.02332745766936583040110129147, 8.806655653872362639529637943363, 9.502657276706303082172623972497, 9.991396882815858988684437517264, 11.90054603684560371513123506662, 13.39178489842652739710856462010, 13.95618726829362024059731865348