| L(s) = 1 | − 2.64·2-s − 25.0·4-s + 25·5-s − 12.8·7-s + 150.·8-s − 66.0·10-s + 121·11-s − 485.·13-s + 34.0·14-s + 402.·16-s − 266.·17-s − 149.·19-s − 625.·20-s − 319.·22-s + 3.21e3·23-s + 625·25-s + 1.28e3·26-s + 322.·28-s − 2.94e3·29-s + 2.14e3·31-s − 5.88e3·32-s + 704.·34-s − 322.·35-s − 808.·37-s + 395.·38-s + 3.76e3·40-s − 1.01e4·41-s + ⋯ |
| L(s) = 1 | − 0.467·2-s − 0.781·4-s + 0.447·5-s − 0.0993·7-s + 0.832·8-s − 0.208·10-s + 0.301·11-s − 0.796·13-s + 0.0464·14-s + 0.393·16-s − 0.223·17-s − 0.0951·19-s − 0.349·20-s − 0.140·22-s + 1.26·23-s + 0.200·25-s + 0.371·26-s + 0.0776·28-s − 0.651·29-s + 0.401·31-s − 1.01·32-s + 0.104·34-s − 0.0444·35-s − 0.0970·37-s + 0.0444·38-s + 0.372·40-s − 0.938·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.197659218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197659218\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 2.64T + 32T^{2} \) |
| 7 | \( 1 + 12.8T + 1.68e4T^{2} \) |
| 13 | \( 1 + 485.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 266.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 149.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.21e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 808.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.50e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.59e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.27e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.941302815387080698791708644253, −9.306373238400890244723016682007, −8.558880208950025727470035861032, −7.53824183034612689736227043631, −6.58764498389755141195601813002, −5.31330703693324128712628949832, −4.56627158326884157866075202468, −3.27863304396712639620309668629, −1.84306785011680142775085767142, −0.60442935246320977603063757196,
0.60442935246320977603063757196, 1.84306785011680142775085767142, 3.27863304396712639620309668629, 4.56627158326884157866075202468, 5.31330703693324128712628949832, 6.58764498389755141195601813002, 7.53824183034612689736227043631, 8.558880208950025727470035861032, 9.306373238400890244723016682007, 9.941302815387080698791708644253
