| L(s) = 1 | + 40.2·2-s + 243·3-s − 429.·4-s + 7.75e3·5-s + 9.77e3·6-s + 4.26e4·7-s − 9.96e4·8-s + 5.90e4·9-s + 3.11e5·10-s + 1.98e5·11-s − 1.04e5·12-s − 1.67e6·13-s + 1.71e6·14-s + 1.88e6·15-s − 3.13e6·16-s − 3.70e6·17-s + 2.37e6·18-s − 2.06e7·19-s − 3.32e6·20-s + 1.03e7·21-s + 7.98e6·22-s − 8.94e6·23-s − 2.42e7·24-s + 1.12e7·25-s − 6.72e7·26-s + 1.43e7·27-s − 1.82e7·28-s + ⋯ |
| L(s) = 1 | + 0.889·2-s + 0.577·3-s − 0.209·4-s + 1.10·5-s + 0.513·6-s + 0.958·7-s − 1.07·8-s + 0.333·9-s + 0.986·10-s + 0.371·11-s − 0.120·12-s − 1.24·13-s + 0.852·14-s + 0.640·15-s − 0.746·16-s − 0.632·17-s + 0.296·18-s − 1.91·19-s − 0.232·20-s + 0.553·21-s + 0.330·22-s − 0.289·23-s − 0.620·24-s + 0.230·25-s − 1.11·26-s + 0.192·27-s − 0.200·28-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 | \( 1 - 40.2T + 2.04e3T^{2} \) |
| 5 | \( 1 - 7.75e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 4.26e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.98e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.67e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.70e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 2.06e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 8.94e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 3.99e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.18e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.76e6T + 1.77e17T^{2} \) |
| 41 | \( 1 - 6.45e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.91e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.30e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.69e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 5.61e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.16e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.23e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.59e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.32e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.72e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.57e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 9.36e10T + 7.15e21T^{2} \) |
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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
−10.02656455499075961404287939680, −9.154192287018676203948781320582, −8.317305535937282314860100280507, −6.87969885738030070856029577020, −5.75302197749767168771143427918, −4.79804065845696392993226577056, −3.97540415668200131972463613495, −2.44247104235441647077859346312, −1.82083742375174775200983468380, 0,
1.82083742375174775200983468380, 2.44247104235441647077859346312, 3.97540415668200131972463613495, 4.79804065845696392993226577056, 5.75302197749767168771143427918, 6.87969885738030070856029577020, 8.317305535937282314860100280507, 9.154192287018676203948781320582, 10.02656455499075961404287939680
