| L(s) = 1 | − 2.23·2-s − 3-s + 3.00·4-s + 1.23·5-s + 2.23·6-s + 3.23·7-s − 2.23·8-s + 9-s − 2.76·10-s + 4·11-s − 3.00·12-s − 4.47·13-s − 7.23·14-s − 1.23·15-s − 0.999·16-s − 2.76·17-s − 2.23·18-s + 7.23·19-s + 3.70·20-s − 3.23·21-s − 8.94·22-s + 23-s + 2.23·24-s − 3.47·25-s + 10.0·26-s − 27-s + 9.70·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 0.577·3-s + 1.50·4-s + 0.552·5-s + 0.912·6-s + 1.22·7-s − 0.790·8-s + 0.333·9-s − 0.874·10-s + 1.20·11-s − 0.866·12-s − 1.24·13-s − 1.93·14-s − 0.319·15-s − 0.249·16-s − 0.670·17-s − 0.527·18-s + 1.66·19-s + 0.829·20-s − 0.706·21-s − 1.90·22-s + 0.208·23-s + 0.456·24-s − 0.694·25-s + 1.96·26-s − 0.192·27-s + 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4725447308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4725447308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 5.23T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 79 | \( 1 - 9.70T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−14.95715533007426163850111930637, −13.81899745768686567473769273524, −11.90798729347742989362181053457, −11.28536519904873953585469610543, −9.998840008183561101707583288020, −9.209997901439462058565944429156, −7.85443001137786965466864754159, −6.76275085319516488596762758674, −4.98567104290795192227696532437, −1.65001680377957710884516373019,
1.65001680377957710884516373019, 4.98567104290795192227696532437, 6.76275085319516488596762758674, 7.85443001137786965466864754159, 9.209997901439462058565944429156, 9.998840008183561101707583288020, 11.28536519904873953585469610543, 11.90798729347742989362181053457, 13.81899745768686567473769273524, 14.95715533007426163850111930637
