| L(s) = 1 | − 5·5-s + 10.2·7-s + 5.26·11-s + 34.5·13-s + 74.8·17-s + 50.8·19-s + 81.4·23-s + 25·25-s + 152.·29-s − 93.9·31-s − 51.3·35-s − 167.·37-s + 18.6·41-s − 133.·43-s − 46.8·47-s − 237.·49-s − 105.·53-s − 26.3·55-s + 77.7·59-s − 48.6·61-s − 172.·65-s + 667.·67-s + 344.·71-s − 1.07e3·73-s + 54·77-s + 420.·79-s + 296.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·7-s + 0.144·11-s + 0.737·13-s + 1.06·17-s + 0.614·19-s + 0.738·23-s + 0.200·25-s + 0.975·29-s − 0.544·31-s − 0.247·35-s − 0.743·37-s + 0.0709·41-s − 0.473·43-s − 0.145·47-s − 0.692·49-s − 0.272·53-s − 0.0645·55-s + 0.171·59-s − 0.102·61-s − 0.329·65-s + 1.21·67-s + 0.575·71-s − 1.72·73-s + 0.0799·77-s + 0.599·79-s + 0.392·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.519935011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.519935011\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 10.2T + 343T^{2} \) |
| 11 | \( 1 - 5.26T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 93.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 18.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 133.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 46.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 105.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 77.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 48.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 667.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 344.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 420.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 296.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 950.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 135.T + 9.12e5T^{2} \) |
| 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
−8.593969597595779978703672677904, −8.002509939881123498738742386696, −7.25569842930559656918256312589, −6.42650558685859691074986842127, −5.43888588295361877471700515707, −4.77809215774106530006665206959, −3.71379968511268226597270475688, −3.03625107145262414978158264309, −1.64130826529606213660269853237, −0.76778229190803822022625682566,
0.76778229190803822022625682566, 1.64130826529606213660269853237, 3.03625107145262414978158264309, 3.71379968511268226597270475688, 4.77809215774106530006665206959, 5.43888588295361877471700515707, 6.42650558685859691074986842127, 7.25569842930559656918256312589, 8.002509939881123498738742386696, 8.593969597595779978703672677904
