| L(s) = 1 | + 1.26·3-s + 11.0·5-s + 5.70·7-s − 25.4·9-s + 59.0·11-s + 88.0·13-s + 13.9·15-s − 113.·17-s + 19·19-s + 7.20·21-s + 40.1·23-s − 3.28·25-s − 66.1·27-s − 66.5·29-s − 248.·31-s + 74.5·33-s + 62.9·35-s − 330.·37-s + 111.·39-s − 172.·41-s + 56.9·43-s − 280.·45-s + 483.·47-s − 310.·49-s − 142.·51-s − 104.·53-s + 651.·55-s + ⋯ |
| L(s) = 1 | + 0.243·3-s + 0.986·5-s + 0.308·7-s − 0.940·9-s + 1.61·11-s + 1.87·13-s + 0.239·15-s − 1.61·17-s + 0.229·19-s + 0.0748·21-s + 0.364·23-s − 0.0263·25-s − 0.471·27-s − 0.426·29-s − 1.44·31-s + 0.393·33-s + 0.303·35-s − 1.47·37-s + 0.456·39-s − 0.655·41-s + 0.201·43-s − 0.928·45-s + 1.49·47-s − 0.905·49-s − 0.392·51-s − 0.270·53-s + 1.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.892626109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892626109\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 1.26T + 27T^{2} \) |
| 5 | \( 1 - 11.0T + 125T^{2} \) |
| 7 | \( 1 - 5.70T + 343T^{2} \) |
| 11 | \( 1 - 59.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 88.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 113.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 40.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 66.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 172.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 56.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 579.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 314.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 12.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 711.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 704.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 50.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 849.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 704.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 232.T + 9.12e5T^{2} \) |
| 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
−13.91767409938634973728453541248, −13.29049409512561326056522709250, −11.62198137775101094774258448984, −10.84304339645499939998125774853, −9.125718656623211975895720929268, −8.718607441677962935349668106990, −6.68582033822393918156009275920, −5.70234457510118832252743648478, −3.75871904875731464056622455498, −1.74181127422057721190169609644,
1.74181127422057721190169609644, 3.75871904875731464056622455498, 5.70234457510118832252743648478, 6.68582033822393918156009275920, 8.718607441677962935349668106990, 9.125718656623211975895720929268, 10.84304339645499939998125774853, 11.62198137775101094774258448984, 13.29049409512561326056522709250, 13.91767409938634973728453541248
