| L(s) = 1 | + 59.7·2-s + 1.52e3·4-s + 6.37e3·5-s − 2.30e4·7-s − 3.13e4·8-s + 3.81e5·10-s − 5.22e5·11-s + 1.47e6·13-s − 1.37e6·14-s − 4.99e6·16-s − 4.61e6·17-s + 1.36e7·19-s + 9.71e6·20-s − 3.12e7·22-s + 5.58e7·23-s − 8.13e6·25-s + 8.84e7·26-s − 3.50e7·28-s − 2.05e7·29-s − 1.47e8·31-s − 2.34e8·32-s − 2.75e8·34-s − 1.46e8·35-s − 1.64e8·37-s + 8.14e8·38-s − 2.00e8·40-s − 9.49e6·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.743·4-s + 0.912·5-s − 0.517·7-s − 0.338·8-s + 1.20·10-s − 0.978·11-s + 1.10·13-s − 0.683·14-s − 1.19·16-s − 0.788·17-s + 1.26·19-s + 0.678·20-s − 1.29·22-s + 1.80·23-s − 0.166·25-s + 1.45·26-s − 0.384·28-s − 0.185·29-s − 0.924·31-s − 1.23·32-s − 1.04·34-s − 0.472·35-s − 0.390·37-s + 1.66·38-s − 0.309·40-s − 0.0127·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 - 59.7T + 2.04e3T^{2} \) |
| 5 | \( 1 - 6.37e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 2.30e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.22e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.47e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 4.61e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.36e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.58e7T + 9.52e14T^{2} \) |
| 31 | \( 1 + 1.47e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.64e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.49e6T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.59e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.65e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.15e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 2.72e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.15e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.36e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 9.15e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.13e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.95e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.56e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.50e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.86e10T + 7.15e21T^{2} \) |
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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
−9.553560433624335749316313119685, −8.814243268436095589324798302270, −7.27681300442442251672425175399, −6.23691892097053231809172754958, −5.54806414125247272691133839046, −4.74139949703511819410443884895, −3.41833442384756593432505620670, −2.76952989088822671927196995670, −1.50007330509683341664509175713, 0,
1.50007330509683341664509175713, 2.76952989088822671927196995670, 3.41833442384756593432505620670, 4.74139949703511819410443884895, 5.54806414125247272691133839046, 6.23691892097053231809172754958, 7.27681300442442251672425175399, 8.814243268436095589324798302270, 9.553560433624335749316313119685
