| L(s) = 1 | + 23.7·2-s − 1.48e3·4-s − 6.64e3·5-s − 6.11e4·7-s − 8.37e4·8-s − 1.57e5·10-s − 9.71e4·11-s + 2.25e6·13-s − 1.44e6·14-s + 1.05e6·16-s + 5.47e6·17-s − 1.36e7·19-s + 9.87e6·20-s − 2.30e6·22-s − 3.78e7·23-s − 4.68e6·25-s + 5.33e7·26-s + 9.08e7·28-s − 2.05e7·29-s + 7.66e7·31-s + 1.96e8·32-s + 1.29e8·34-s + 4.06e8·35-s + 5.87e8·37-s − 3.24e8·38-s + 5.56e8·40-s + 5.25e8·41-s + ⋯ |
| L(s) = 1 | + 0.523·2-s − 0.725·4-s − 0.950·5-s − 1.37·7-s − 0.903·8-s − 0.497·10-s − 0.181·11-s + 1.68·13-s − 0.720·14-s + 0.252·16-s + 0.934·17-s − 1.26·19-s + 0.689·20-s − 0.0952·22-s − 1.22·23-s − 0.0960·25-s + 0.881·26-s + 0.997·28-s − 0.185·29-s + 0.480·31-s + 1.03·32-s + 0.489·34-s + 1.30·35-s + 1.39·37-s − 0.664·38-s + 0.859·40-s + 0.707·41-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 | \( 1 - 23.7T + 2.04e3T^{2} \) |
| 5 | \( 1 + 6.64e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.11e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.71e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.25e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.47e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.36e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.78e7T + 9.52e14T^{2} \) |
| 31 | \( 1 - 7.66e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.87e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.25e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.40e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.05e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.82e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.81e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.42e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.58e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 7.93e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.06e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.33e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.80e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.03e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 2.92e10T + 7.15e21T^{2} \) |
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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
−9.540621054179367550932505603446, −8.589719866553230867325835829191, −7.78920065067619458054069109489, −6.29696749549146108939319543852, −5.79551771274150589677577173570, −4.05960021400307893514514599199, −3.88849401459642298286274370906, −2.77152463435276191965264024632, −0.863227099518990276052505914938, 0,
0.863227099518990276052505914938, 2.77152463435276191965264024632, 3.88849401459642298286274370906, 4.05960021400307893514514599199, 5.79551771274150589677577173570, 6.29696749549146108939319543852, 7.78920065067619458054069109489, 8.589719866553230867325835829191, 9.540621054179367550932505603446
