| L(s) = 1 | + 0.635·2-s + 3-s − 1.59·4-s + 5-s + 0.635·6-s + 4.78·7-s − 2.28·8-s + 9-s + 0.635·10-s + 0.276·11-s − 1.59·12-s + 4.93·13-s + 3.04·14-s + 15-s + 1.73·16-s + 2.88·17-s + 0.635·18-s − 4.56·19-s − 1.59·20-s + 4.78·21-s + 0.175·22-s − 2.28·24-s + 25-s + 3.13·26-s + 27-s − 7.63·28-s + 3.42·29-s + ⋯ |
| L(s) = 1 | + 0.449·2-s + 0.577·3-s − 0.797·4-s + 0.447·5-s + 0.259·6-s + 1.80·7-s − 0.808·8-s + 0.333·9-s + 0.201·10-s + 0.0832·11-s − 0.460·12-s + 1.36·13-s + 0.812·14-s + 0.258·15-s + 0.434·16-s + 0.700·17-s + 0.149·18-s − 1.04·19-s − 0.356·20-s + 1.04·21-s + 0.0374·22-s − 0.466·24-s + 0.200·25-s + 0.615·26-s + 0.192·27-s − 1.44·28-s + 0.635·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.123457880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.123457880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.635T + 2T^{2} \) |
| 7 | \( 1 - 4.78T + 7T^{2} \) |
| 11 | \( 1 - 0.276T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 - 2.88T + 17T^{2} \) |
| 19 | \( 1 + 4.56T + 19T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 - 4.64T + 37T^{2} \) |
| 41 | \( 1 - 1.66T + 41T^{2} \) |
| 43 | \( 1 + 0.886T + 43T^{2} \) |
| 47 | \( 1 + 13.5T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 6.87T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 9.69T + 89T^{2} \) |
| 97 | \( 1 + 0.0430T + 97T^{2} \) |
| 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
−8.122480840633753674480763513233, −7.28831232146518078680231714478, −6.17839798111434072021740542985, −5.73881725331082220751490965794, −4.78479291927298249539885215285, −4.43903848241567478896347422735, −3.66458672888546799527490579479, −2.76061798206151410091180656775, −1.71202644302847303231104054620, −1.02774611201056359424203408121,
1.02774611201056359424203408121, 1.71202644302847303231104054620, 2.76061798206151410091180656775, 3.66458672888546799527490579479, 4.43903848241567478896347422735, 4.78479291927298249539885215285, 5.73881725331082220751490965794, 6.17839798111434072021740542985, 7.28831232146518078680231714478, 8.122480840633753674480763513233
