L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 7.24·5-s + 6·6-s + 30.2·7-s + 8·8-s + 9·9-s + 14.4·10-s + 47.4·11-s + 12·12-s + 70.6·13-s + 60.4·14-s + 21.7·15-s + 16·16-s − 29.4·17-s + 18·18-s + 28.9·20-s + 90.6·21-s + 94.8·22-s − 43.0·23-s + 24·24-s − 72.4·25-s + 141.·26-s + 27·27-s + 120.·28-s + 15.7·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.648·5-s + 0.408·6-s + 1.63·7-s + 0.353·8-s + 0.333·9-s + 0.458·10-s + 1.29·11-s + 0.288·12-s + 1.50·13-s + 1.15·14-s + 0.374·15-s + 0.250·16-s − 0.419·17-s + 0.235·18-s + 0.324·20-s + 0.941·21-s + 0.919·22-s − 0.389·23-s + 0.204·24-s − 0.579·25-s + 1.06·26-s + 0.192·27-s + 0.815·28-s + 0.101·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(7.853796767\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.853796767\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 7.24T + 125T^{2} \) |
| 7 | \( 1 - 30.2T + 343T^{2} \) |
| 11 | \( 1 - 47.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 43.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 15.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 291.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 18.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 234.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 264.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 343.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 718.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 840.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 898.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 904.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 212.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 499.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 784.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 60.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 889.T + 9.12e5T^{2} \) |
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
−8.702047439936899973286401318617, −8.010006851696793577658623203526, −7.11243464659254377459393289911, −6.18561703838415038060918090909, −5.62087088034140836336295156653, −4.40783338031470150506760061686, −4.08100574904459276470432531672, −2.88435458846259917307062202783, −1.63817157372709106295748638178, −1.41574074910932472181746502787,
1.41574074910932472181746502787, 1.63817157372709106295748638178, 2.88435458846259917307062202783, 4.08100574904459276470432531672, 4.40783338031470150506760061686, 5.62087088034140836336295156653, 6.18561703838415038060918090909, 7.11243464659254377459393289911, 8.010006851696793577658623203526, 8.702047439936899973286401318617