| L(s) = 1 | − 3.21·3-s + 5·5-s + 21.1·7-s − 16.6·9-s + 6.43·11-s + 34.7·13-s − 16.0·15-s + 65.5·17-s − 20.9·19-s − 68.0·21-s − 71.5·23-s + 25·25-s + 140.·27-s − 29·29-s + 268.·31-s − 20.7·33-s + 105.·35-s + 241.·37-s − 111.·39-s − 299.·41-s − 280.·43-s − 83.1·45-s + 20.2·47-s + 103.·49-s − 211.·51-s + 260.·53-s + 32.1·55-s + ⋯ |
| L(s) = 1 | − 0.619·3-s + 0.447·5-s + 1.14·7-s − 0.616·9-s + 0.176·11-s + 0.740·13-s − 0.277·15-s + 0.935·17-s − 0.253·19-s − 0.706·21-s − 0.649·23-s + 0.200·25-s + 1.00·27-s − 0.185·29-s + 1.55·31-s − 0.109·33-s + 0.510·35-s + 1.07·37-s − 0.458·39-s − 1.14·41-s − 0.995·43-s − 0.275·45-s + 0.0629·47-s + 0.301·49-s − 0.579·51-s + 0.674·53-s + 0.0789·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.153097758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153097758\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 3.21T + 27T^{2} \) |
| 7 | \( 1 - 21.1T + 343T^{2} \) |
| 11 | \( 1 - 6.43T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 71.5T + 1.21e4T^{2} \) |
| 31 | \( 1 - 268.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 280.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 20.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 260.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 163.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 487.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 203.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 118.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 792.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 129.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 587.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 134.T + 9.12e5T^{2} \) |
| 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
−9.479789103901307593992480898884, −8.344666625154221093841397688718, −8.063234374279017186090145423238, −6.70616274943825166261096591261, −5.95798474611964789825842095749, −5.25616683452598389696444275782, −4.39941259220737002414647050011, −3.13001290110558483154949466007, −1.84373705978565891534082909518, −0.816534940151619076487862785072,
0.816534940151619076487862785072, 1.84373705978565891534082909518, 3.13001290110558483154949466007, 4.39941259220737002414647050011, 5.25616683452598389696444275782, 5.95798474611964789825842095749, 6.70616274943825166261096591261, 8.063234374279017186090145423238, 8.344666625154221093841397688718, 9.479789103901307593992480898884
