| L(s) = 1 | + 2.39·3-s + 5-s + 4.65·7-s + 2.73·9-s + 6.03·13-s + 2.39·15-s + 3.32·17-s − 2.73·19-s + 11.1·21-s − 1.97·23-s + 25-s − 0.626·27-s − 2.29·29-s − 6.95·31-s + 4.65·35-s − 3.83·37-s + 14.4·39-s − 3.79·41-s − 7.98·43-s + 2.73·45-s − 12.1·47-s + 14.6·49-s + 7.95·51-s − 11.5·53-s − 6.56·57-s + 7.09·59-s + 1.33·61-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 0.447·5-s + 1.75·7-s + 0.912·9-s + 1.67·13-s + 0.618·15-s + 0.805·17-s − 0.628·19-s + 2.43·21-s − 0.412·23-s + 0.200·25-s − 0.120·27-s − 0.425·29-s − 1.24·31-s + 0.786·35-s − 0.630·37-s + 2.31·39-s − 0.592·41-s − 1.21·43-s + 0.408·45-s − 1.77·47-s + 2.09·49-s + 1.11·51-s − 1.57·53-s − 0.868·57-s + 0.924·59-s + 0.171·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.096251613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.096251613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 13 | \( 1 - 6.03T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 + 1.97T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 + 3.83T + 37T^{2} \) |
| 41 | \( 1 + 3.79T + 41T^{2} \) |
| 43 | \( 1 + 7.98T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 + 7.18T + 67T^{2} \) |
| 71 | \( 1 - 6.74T + 71T^{2} \) |
| 73 | \( 1 - 8.71T + 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 - 8.21T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.54T + 97T^{2} \) |
| 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
−8.718686720311200840982208864625, −8.224871811102919337152804791419, −7.86115452783854530042278077837, −6.77750925544897183562512982453, −5.73688765189083705439889422307, −4.95706341729059692870129779038, −3.90056837456233010124404700847, −3.25795044719271829915496650443, −1.88684894907553735645733453195, −1.56928576846308433924409817201,
1.56928576846308433924409817201, 1.88684894907553735645733453195, 3.25795044719271829915496650443, 3.90056837456233010124404700847, 4.95706341729059692870129779038, 5.73688765189083705439889422307, 6.77750925544897183562512982453, 7.86115452783854530042278077837, 8.224871811102919337152804791419, 8.718686720311200840982208864625
