| L(s) = 1 | − 3-s + 5-s − 3.22·7-s + 9-s − 2·11-s − 3.22·13-s − 15-s + 6.88·17-s − 4·19-s + 3.22·21-s + 6.88·23-s + 25-s − 27-s − 1.65·29-s + 31-s + 2·33-s − 3.22·35-s − 5.22·37-s + 3.22·39-s + 4·41-s − 2·43-s + 45-s − 1.57·47-s + 3.42·49-s − 6.88·51-s + 0.425·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.22·7-s + 0.333·9-s − 0.603·11-s − 0.895·13-s − 0.258·15-s + 1.66·17-s − 0.917·19-s + 0.704·21-s + 1.43·23-s + 0.200·25-s − 0.192·27-s − 0.307·29-s + 0.179·31-s + 0.348·33-s − 0.545·35-s − 0.859·37-s + 0.517·39-s + 0.624·41-s − 0.304·43-s + 0.149·45-s − 0.229·47-s + 0.489·49-s − 0.963·51-s + 0.0584·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.043941246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043941246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 - 6.88T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 - 0.425T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 4.80T + 71T^{2} \) |
| 73 | \( 1 + 7.68T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.88T + 83T^{2} \) |
| 89 | \( 1 - 1.88T + 89T^{2} \) |
| 97 | \( 1 + 9.76T + 97T^{2} \) |
| 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
−7.62777253418905463919601920478, −7.21496335261636742721014116182, −6.38348145431119371657913563509, −5.86592016692915793465715288050, −5.16452024356310208219014001132, −4.52022567836723457981312935291, −3.30880918631133960698844545108, −2.89217084999232248716377481333, −1.73439341764415316708003545926, −0.51910176755652387303574761575,
0.51910176755652387303574761575, 1.73439341764415316708003545926, 2.89217084999232248716377481333, 3.30880918631133960698844545108, 4.52022567836723457981312935291, 5.16452024356310208219014001132, 5.86592016692915793465715288050, 6.38348145431119371657913563509, 7.21496335261636742721014116182, 7.62777253418905463919601920478
