| L(s) = 1 | + 75.6·2-s + 243·3-s + 3.67e3·4-s − 5.74e3·5-s + 1.83e4·6-s + 1.09e4·7-s + 1.22e5·8-s + 5.90e4·9-s − 4.34e5·10-s − 3.43e5·11-s + 8.92e5·12-s + 1.01e6·13-s + 8.27e5·14-s − 1.39e6·15-s + 1.77e6·16-s − 6.50e6·17-s + 4.46e6·18-s − 2.12e7·19-s − 2.10e7·20-s + 2.65e6·21-s − 2.59e7·22-s + 8.47e6·23-s + 2.98e7·24-s − 1.58e7·25-s + 7.67e7·26-s + 1.43e7·27-s + 4.01e7·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 0.577·3-s + 1.79·4-s − 0.821·5-s + 0.965·6-s + 0.246·7-s + 1.32·8-s + 0.333·9-s − 1.37·10-s − 0.642·11-s + 1.03·12-s + 0.757·13-s + 0.411·14-s − 0.474·15-s + 0.424·16-s − 1.11·17-s + 0.557·18-s − 1.96·19-s − 1.47·20-s + 0.142·21-s − 1.07·22-s + 0.274·23-s + 0.766·24-s − 0.325·25-s + 1.26·26-s + 0.192·27-s + 0.441·28-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 | \( 1 - 75.6T + 2.04e3T^{2} \) |
| 5 | \( 1 + 5.74e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 1.09e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 3.43e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.01e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.50e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 2.12e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 8.47e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 7.43e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 6.43e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.96e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.07e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.20e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.66e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.12e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 1.12e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.98e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.52e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.92e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.62e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.51e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 3.09e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 9.89e10T + 7.15e21T^{2} \) |
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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
−10.74980852602913225031284943180, −8.922424737105785610237429974854, −7.960938034557753569112773092051, −6.85132240581884021453426782646, −5.84052221358806383579697131539, −4.45570207367355980701594895689, −4.03343371064337642146130386425, −2.85355193395909264017083350580, −1.91172462930712744546163525501, 0,
1.91172462930712744546163525501, 2.85355193395909264017083350580, 4.03343371064337642146130386425, 4.45570207367355980701594895689, 5.84052221358806383579697131539, 6.85132240581884021453426782646, 7.960938034557753569112773092051, 8.922424737105785610237429974854, 10.74980852602913225031284943180
