| L(s) = 1 | + 81.6·2-s + 243·3-s + 4.61e3·4-s + 4.43e3·5-s + 1.98e4·6-s − 6.76e4·7-s + 2.09e5·8-s + 5.90e4·9-s + 3.61e5·10-s − 5.19e5·11-s + 1.12e6·12-s − 2.46e6·13-s − 5.52e6·14-s + 1.07e6·15-s + 7.64e6·16-s − 4.35e6·17-s + 4.81e6·18-s + 3.11e6·19-s + 2.04e7·20-s − 1.64e7·21-s − 4.24e7·22-s − 1.04e7·23-s + 5.08e7·24-s − 2.92e7·25-s − 2.00e8·26-s + 1.43e7·27-s − 3.12e8·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.577·3-s + 2.25·4-s + 0.633·5-s + 1.04·6-s − 1.52·7-s + 2.25·8-s + 0.333·9-s + 1.14·10-s − 0.973·11-s + 1.30·12-s − 1.83·13-s − 2.74·14-s + 0.366·15-s + 1.82·16-s − 0.743·17-s + 0.601·18-s + 0.288·19-s + 1.42·20-s − 0.878·21-s − 1.75·22-s − 0.338·23-s + 1.30·24-s − 0.598·25-s − 3.31·26-s + 0.192·27-s − 3.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 243T \) |
| 59 | \( 1 + 7.14e8T \) |
| good | 2 | \( 1 - 81.6T + 2.04e3T^{2} \) |
| 5 | \( 1 - 4.43e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.76e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.19e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.46e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 4.35e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 3.11e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.04e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.13e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.99e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 7.90e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 4.32e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 4.93e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 3.47e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 6.14e8T + 9.26e18T^{2} \) |
| 61 | \( 1 - 9.09e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.20e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.45e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 8.48e8T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.82e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.31e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.16e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.63e10T + 7.15e21T^{2} \) |
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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
−10.09014138432589865619867390048, −9.565732618650075834269991314595, −7.65970373984780150716526739339, −6.75810095305038444769142678008, −5.82310203516868487709833595964, −4.86375626731128711756181231813, −3.71041225816673383631852555526, −2.63428524620742245389387246781, −2.25217164055018597442110191308, 0,
2.25217164055018597442110191308, 2.63428524620742245389387246781, 3.71041225816673383631852555526, 4.86375626731128711756181231813, 5.82310203516868487709833595964, 6.75810095305038444769142678008, 7.65970373984780150716526739339, 9.565732618650075834269991314595, 10.09014138432589865619867390048
