| L(s) = 1 | + 3·3-s − 9.38·5-s − 32.1·7-s + 9·9-s + 48.7·11-s − 13·13-s − 28.1·15-s − 110.·17-s + 136.·19-s − 96.4·21-s + 32.9·23-s − 37·25-s + 27·27-s + 133.·29-s − 180.·31-s + 146.·33-s + 301.·35-s − 114.·37-s − 39·39-s + 116.·41-s + 502.·43-s − 84.4·45-s + 526.·47-s + 690.·49-s − 330.·51-s + 382.·53-s − 457.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.839·5-s − 1.73·7-s + 0.333·9-s + 1.33·11-s − 0.277·13-s − 0.484·15-s − 1.57·17-s + 1.64·19-s − 1.00·21-s + 0.298·23-s − 0.295·25-s + 0.192·27-s + 0.855·29-s − 1.04·31-s + 0.771·33-s + 1.45·35-s − 0.510·37-s − 0.160·39-s + 0.444·41-s + 1.78·43-s − 0.279·45-s + 1.63·47-s + 2.01·49-s − 0.908·51-s + 0.991·53-s − 1.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.576020226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576020226\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 9.38T + 125T^{2} \) |
| 7 | \( 1 + 32.1T + 343T^{2} \) |
| 11 | \( 1 - 48.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 32.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 180.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 116.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 526.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 382.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 801.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 266.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 213.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 655.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 766.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 314.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 730.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 20.5T + 9.12e5T^{2} \) |
| 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
−9.957159283065952668478780352958, −9.238148744477759676165005157610, −8.739905847972554722664849207605, −7.27266472131580327233242162004, −6.94393588882421092131409481161, −5.81232623992450512103764085810, −4.18599566329695938274823370100, −3.61057417857862815454353461596, −2.55510105406401119580967035155, −0.70888115767307727695894001009,
0.70888115767307727695894001009, 2.55510105406401119580967035155, 3.61057417857862815454353461596, 4.18599566329695938274823370100, 5.81232623992450512103764085810, 6.94393588882421092131409481161, 7.27266472131580327233242162004, 8.739905847972554722664849207605, 9.238148744477759676165005157610, 9.957159283065952668478780352958
