| L(s) = 1 | − 3.59·2-s + 3·3-s + 4.89·4-s − 10.7·6-s + 16.1·7-s + 11.1·8-s + 9·9-s + 11·11-s + 14.6·12-s + 54.1·13-s − 57.9·14-s − 79.2·16-s + 107.·17-s − 32.3·18-s + 48.7·19-s + 48.4·21-s − 39.4·22-s − 11.9·23-s + 33.4·24-s − 194.·26-s + 27·27-s + 78.9·28-s + 239.·29-s − 82.0·31-s + 195.·32-s + 33·33-s − 384.·34-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.577·3-s + 0.611·4-s − 0.732·6-s + 0.871·7-s + 0.493·8-s + 0.333·9-s + 0.301·11-s + 0.353·12-s + 1.15·13-s − 1.10·14-s − 1.23·16-s + 1.52·17-s − 0.423·18-s + 0.588·19-s + 0.503·21-s − 0.382·22-s − 0.108·23-s + 0.284·24-s − 1.46·26-s + 0.192·27-s + 0.533·28-s + 1.53·29-s − 0.475·31-s + 1.07·32-s + 0.174·33-s − 1.93·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.737398709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737398709\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 3.59T + 8T^{2} \) |
| 7 | \( 1 - 16.1T + 343T^{2} \) |
| 13 | \( 1 - 54.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 11.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 239.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 21.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 124.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 233.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 232.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 876.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 733.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 588.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.54e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.784332700739636120047871256374, −8.824538265957090862984442657588, −8.274669889890678787790145893236, −7.70231136854552248755973304148, −6.74517118727960106577098322585, −5.45186028111385934401011142552, −4.31951304257228794079168632437, −3.17039257360102443364282472464, −1.65470999538928771380172920643, −0.978278863960150269743198770832,
0.978278863960150269743198770832, 1.65470999538928771380172920643, 3.17039257360102443364282472464, 4.31951304257228794079168632437, 5.45186028111385934401011142552, 6.74517118727960106577098322585, 7.70231136854552248755973304148, 8.274669889890678787790145893236, 8.824538265957090862984442657588, 9.784332700739636120047871256374
