| L(s) = 1 | − 5·5-s + 9.46·7-s − 49.0·11-s + 55.4·13-s + 27.9·17-s + 26.3·19-s − 9.53·23-s + 25·25-s − 218.·29-s − 158.·31-s − 47.3·35-s + 189.·37-s + 246.·41-s + 40.2·43-s + 213.·47-s − 253.·49-s + 283.·53-s + 245.·55-s + 92·59-s + 370.·61-s − 277.·65-s + 1.08e3·67-s − 333.·71-s − 606.·73-s − 464.·77-s − 44.3·79-s + 107.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.511·7-s − 1.34·11-s + 1.18·13-s + 0.399·17-s + 0.318·19-s − 0.0864·23-s + 0.200·25-s − 1.39·29-s − 0.920·31-s − 0.228·35-s + 0.842·37-s + 0.937·41-s + 0.142·43-s + 0.664·47-s − 0.738·49-s + 0.734·53-s + 0.601·55-s + 0.203·59-s + 0.776·61-s − 0.528·65-s + 1.98·67-s − 0.557·71-s − 0.972·73-s − 0.687·77-s − 0.0631·79-s + 0.142·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.821739246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821739246\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 9.46T + 343T^{2} \) |
| 11 | \( 1 + 49.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.53T + 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 246.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 40.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 213.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92T + 2.05e5T^{2} \) |
| 61 | \( 1 - 370.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.08e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 606.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 44.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 107.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 257.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
| 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
−9.463610380468137059124679827827, −8.537774972035303840631030684562, −7.83950516558325037752876989357, −7.24376618121188601251448817185, −5.91508107824151688638658826860, −5.29806262730187595240166070137, −4.17913637251256716492393273133, −3.27158258157935411961709577307, −2.04844775726154793262508172964, −0.70781745139631214253111536534,
0.70781745139631214253111536534, 2.04844775726154793262508172964, 3.27158258157935411961709577307, 4.17913637251256716492393273133, 5.29806262730187595240166070137, 5.91508107824151688638658826860, 7.24376618121188601251448817185, 7.83950516558325037752876989357, 8.537774972035303840631030684562, 9.463610380468137059124679827827
