| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 14.8·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 29.7·10-s + 26.4·11-s − 12·12-s + 59.8·13-s + 14·14-s + 44.5·15-s + 16·16-s − 105.·17-s − 18·18-s − 19·19-s − 59.4·20-s + 21·21-s − 52.8·22-s − 131.·23-s + 24·24-s + 95.5·25-s − 119.·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.32·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.939·10-s + 0.724·11-s − 0.288·12-s + 1.27·13-s + 0.267·14-s + 0.766·15-s + 0.250·16-s − 1.49·17-s − 0.235·18-s − 0.229·19-s − 0.664·20-s + 0.218·21-s − 0.512·22-s − 1.19·23-s + 0.204·24-s + 0.764·25-s − 0.902·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4524091378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4524091378\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 + 14.8T + 125T^{2} \) |
| 11 | \( 1 - 26.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 166.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 79.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 194.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 385.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 544.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 445.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 677.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 473.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 885.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 700.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 323.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 426.T + 9.12e5T^{2} \) |
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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
−9.952915159201415653957971788871, −8.763494390941736322644527842103, −8.422735464382300487868475052077, −7.18510180781921150825124283204, −6.66493717069866220047276064301, −5.66194986954338379797097652137, −4.13779498580334192344891329229, −3.63580281844221035020226840060, −1.85217993873232884664155557182, −0.42419899151753311548138353883,
0.42419899151753311548138353883, 1.85217993873232884664155557182, 3.63580281844221035020226840060, 4.13779498580334192344891329229, 5.66194986954338379797097652137, 6.66493717069866220047276064301, 7.18510180781921150825124283204, 8.422735464382300487868475052077, 8.763494390941736322644527842103, 9.952915159201415653957971788871
