| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 79.8·5-s + 36·6-s + 56.6·7-s − 64·8-s + 81·9-s − 319.·10-s + 280.·11-s − 144·12-s + 515.·13-s − 226.·14-s − 719.·15-s + 256·16-s + 775.·17-s − 324·18-s + 2.57e3·19-s + 1.27e3·20-s − 509.·21-s − 1.12e3·22-s + 2.13e3·23-s + 576·24-s + 3.25e3·25-s − 2.06e3·26-s − 729·27-s + 906.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.42·5-s + 0.408·6-s + 0.436·7-s − 0.353·8-s + 0.333·9-s − 1.01·10-s + 0.698·11-s − 0.288·12-s + 0.845·13-s − 0.308·14-s − 0.825·15-s + 0.250·16-s + 0.650·17-s − 0.235·18-s + 1.63·19-s + 0.714·20-s − 0.252·21-s − 0.493·22-s + 0.840·23-s + 0.204·24-s + 1.04·25-s − 0.598·26-s − 0.192·27-s + 0.218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.147132765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147132765\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 5 | \( 1 - 79.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 56.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 280.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 515.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 775.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.13e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 712.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.14e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.93e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.26e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.35e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
| 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
−10.58698196380694534689690437303, −9.602983708061770776894819925215, −9.133625341377125599430790950940, −7.83090004483877559727690586690, −6.72784744010980329549330297010, −5.88146316548417828154172137885, −5.07789283855646793077848986318, −3.28729048793884930325793374234, −1.69616157155046578130593369696, −1.02816397757589297201835643619,
1.02816397757589297201835643619, 1.69616157155046578130593369696, 3.28729048793884930325793374234, 5.07789283855646793077848986318, 5.88146316548417828154172137885, 6.72784744010980329549330297010, 7.83090004483877559727690586690, 9.133625341377125599430790950940, 9.602983708061770776894819925215, 10.58698196380694534689690437303
