| L(s) = 1 | − 2.07·2-s + 2.30·4-s − 3.91·5-s − 3.22·7-s − 0.627·8-s + 8.12·10-s + 3.34·11-s + 1.97·13-s + 6.69·14-s − 3.30·16-s + 5.04·17-s − 19-s − 9.01·20-s − 6.94·22-s + 5.87·23-s + 10.3·25-s − 4.09·26-s − 7.43·28-s + 5.27·29-s + 4.59·31-s + 8.10·32-s − 10.4·34-s + 12.6·35-s − 4.20·37-s + 2.07·38-s + 2.45·40-s + 5.09·41-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.15·4-s − 1.75·5-s − 1.22·7-s − 0.222·8-s + 2.56·10-s + 1.00·11-s + 0.547·13-s + 1.78·14-s − 0.825·16-s + 1.22·17-s − 0.229·19-s − 2.01·20-s − 1.48·22-s + 1.22·23-s + 2.06·25-s − 0.803·26-s − 1.40·28-s + 0.979·29-s + 0.824·31-s + 1.43·32-s − 1.79·34-s + 2.13·35-s − 0.691·37-s + 0.336·38-s + 0.388·40-s + 0.796·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6439899459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6439899459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 2.07T + 2T^{2} \) |
| 5 | \( 1 + 3.91T + 5T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 23 | \( 1 - 5.87T + 23T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 - 6.90T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 - 5.63T + 67T^{2} \) |
| 71 | \( 1 - 5.18T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 + 2.05T + 79T^{2} \) |
| 83 | \( 1 - 1.89T + 83T^{2} \) |
| 89 | \( 1 - 3.76T + 89T^{2} \) |
| 97 | \( 1 + 2.23T + 97T^{2} \) |
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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
−8.023603580232497352727737141376, −7.19587517838630954573764652506, −6.86201824875795928767180354367, −6.17591457128347499469096443067, −4.92829548331574464697781067486, −4.00520093454973764370607209184, −3.49160785190079829031197150367, −2.70168606507430390242641514310, −1.10551031681072383047734111141, −0.63166582965664749237344860412,
0.63166582965664749237344860412, 1.10551031681072383047734111141, 2.70168606507430390242641514310, 3.49160785190079829031197150367, 4.00520093454973764370607209184, 4.92829548331574464697781067486, 6.17591457128347499469096443067, 6.86201824875795928767180354367, 7.19587517838630954573764652506, 8.023603580232497352727737141376
