| L(s) = 1 | − 2.73·2-s + 5.46·4-s + 5-s + 7-s − 9.46·8-s − 2.73·10-s + 11-s + 6.19·13-s − 2.73·14-s + 14.9·16-s − 0.464·17-s − 8.46·19-s + 5.46·20-s − 2.73·22-s − 1.53·23-s + 25-s − 16.9·26-s + 5.46·28-s − 2.26·29-s − 8.73·31-s − 21.8·32-s + 1.26·34-s + 35-s + 6.19·37-s + 23.1·38-s − 9.46·40-s − 9.66·41-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 2.73·4-s + 0.447·5-s + 0.377·7-s − 3.34·8-s − 0.863·10-s + 0.301·11-s + 1.71·13-s − 0.730·14-s + 3.73·16-s − 0.112·17-s − 1.94·19-s + 1.22·20-s − 0.582·22-s − 0.320·23-s + 0.200·25-s − 3.31·26-s + 1.03·28-s − 0.421·29-s − 1.56·31-s − 3.86·32-s + 0.217·34-s + 0.169·35-s + 1.01·37-s + 3.75·38-s − 1.49·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 13 | \( 1 - 6.19T + 13T^{2} \) |
| 17 | \( 1 + 0.464T + 17T^{2} \) |
| 19 | \( 1 + 8.46T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 + 2.26T + 29T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 - 6.19T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 - 2.46T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 + 8.92T + 67T^{2} \) |
| 71 | \( 1 - 9.66T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 4.46T + 83T^{2} \) |
| 89 | \( 1 - 2.66T + 89T^{2} \) |
| 97 | \( 1 + 5.73T + 97T^{2} \) |
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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
−8.450769243367842638330331465685, −7.78382536356182733500331059295, −6.83288090840068274112315226912, −6.27874219302585509265440313839, −5.69578107279564780223258428544, −4.16716929164566352906039846776, −3.10683311182927144684045958711, −1.89363901214474185200929657476, −1.47512616450842184737927867608, 0,
1.47512616450842184737927867608, 1.89363901214474185200929657476, 3.10683311182927144684045958711, 4.16716929164566352906039846776, 5.69578107279564780223258428544, 6.27874219302585509265440313839, 6.83288090840068274112315226912, 7.78382536356182733500331059295, 8.450769243367842638330331465685
