| L(s) = 1 | + 2.49·3-s − 15.7·5-s + 30.5·7-s − 20.7·9-s + 39.5·11-s + 80.7·13-s − 39.4·15-s + 74.9·17-s + 19·19-s + 76.3·21-s − 54.9·23-s + 124.·25-s − 119.·27-s − 105.·29-s + 245.·31-s + 98.7·33-s − 483.·35-s − 167.·37-s + 201.·39-s + 433.·41-s + 356.·43-s + 328.·45-s − 403.·47-s + 592.·49-s + 187.·51-s − 489.·53-s − 624.·55-s + ⋯ |
| L(s) = 1 | + 0.480·3-s − 1.41·5-s + 1.65·7-s − 0.769·9-s + 1.08·11-s + 1.72·13-s − 0.678·15-s + 1.06·17-s + 0.229·19-s + 0.793·21-s − 0.498·23-s + 0.995·25-s − 0.849·27-s − 0.677·29-s + 1.42·31-s + 0.520·33-s − 2.33·35-s − 0.744·37-s + 0.827·39-s + 1.65·41-s + 1.26·43-s + 1.08·45-s − 1.25·47-s + 1.72·49-s + 0.513·51-s − 1.26·53-s − 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.923902335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923902335\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 2.49T + 27T^{2} \) |
| 5 | \( 1 + 15.7T + 125T^{2} \) |
| 7 | \( 1 - 30.5T + 343T^{2} \) |
| 11 | \( 1 - 39.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 80.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.9T + 4.91e3T^{2} \) |
| 23 | \( 1 + 54.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 105.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 433.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 403.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 489.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 8.75T + 2.05e5T^{2} \) |
| 61 | \( 1 + 801.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 917.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 104.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 813.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 392.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 122.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 15.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 637.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.17419542458191591203194712287, −11.47121103420999147708438376728, −10.95351903580990817378877306323, −9.057801031034329526910479566503, −8.178868733903636350672506839119, −7.72956482801365984578926553155, −5.96657646036491681795868386982, −4.38415597879893249874246086353, −3.43194784054642256597489380199, −1.27101453924820599031212231835,
1.27101453924820599031212231835, 3.43194784054642256597489380199, 4.38415597879893249874246086353, 5.96657646036491681795868386982, 7.72956482801365984578926553155, 8.178868733903636350672506839119, 9.057801031034329526910479566503, 10.95351903580990817378877306323, 11.47121103420999147708438376728, 12.17419542458191591203194712287
