| L(s) = 1 | + 2.18·2-s − 3.20·4-s − 7.60·5-s + 7·7-s − 24.5·8-s − 16.6·10-s − 11·11-s + 0.174·13-s + 15.3·14-s − 28.0·16-s − 128.·17-s + 141.·19-s + 24.4·20-s − 24.0·22-s + 133.·23-s − 67.1·25-s + 0.381·26-s − 22.4·28-s + 177.·29-s + 48.2·31-s + 134.·32-s − 282.·34-s − 53.2·35-s + 161.·37-s + 310.·38-s + 186.·40-s + 195.·41-s + ⋯ |
| L(s) = 1 | + 0.773·2-s − 0.401·4-s − 0.680·5-s + 0.377·7-s − 1.08·8-s − 0.526·10-s − 0.301·11-s + 0.00371·13-s + 0.292·14-s − 0.438·16-s − 1.83·17-s + 1.71·19-s + 0.272·20-s − 0.233·22-s + 1.20·23-s − 0.537·25-s + 0.00287·26-s − 0.151·28-s + 1.13·29-s + 0.279·31-s + 0.745·32-s − 1.42·34-s − 0.257·35-s + 0.718·37-s + 1.32·38-s + 0.737·40-s + 0.745·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.895703419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895703419\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 2.18T + 8T^{2} \) |
| 5 | \( 1 + 7.60T + 125T^{2} \) |
| 13 | \( 1 - 0.174T + 2.19e3T^{2} \) |
| 17 | \( 1 + 128.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 133.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 177.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 48.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 488.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 171.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 431.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 194.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 585.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 155.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 374.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 210.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 7.00T + 4.93e5T^{2} \) |
| 83 | \( 1 - 93.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 307.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 965.T + 9.12e5T^{2} \) |
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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
−10.05533273953525574010816259034, −9.083977734661292411726894512078, −8.391614100162674214411415807073, −7.39825146645164854647279020770, −6.41434031380007236989548736861, −5.21455232076949395584825089619, −4.61856748588601953014376286832, −3.65767286324615624730066230898, −2.60010869480101745261437697292, −0.70483275033689325434338001405,
0.70483275033689325434338001405, 2.60010869480101745261437697292, 3.65767286324615624730066230898, 4.61856748588601953014376286832, 5.21455232076949395584825089619, 6.41434031380007236989548736861, 7.39825146645164854647279020770, 8.391614100162674214411415807073, 9.083977734661292411726894512078, 10.05533273953525574010816259034
