| L(s) = 1 | + 10.6·2-s + 243·3-s − 1.93e3·4-s + 1.26e4·5-s + 2.59e3·6-s − 7.56e4·7-s − 4.25e4·8-s + 5.90e4·9-s + 1.34e5·10-s + 7.75e4·11-s − 4.69e5·12-s + 1.47e6·13-s − 8.09e5·14-s + 3.06e6·15-s + 3.50e6·16-s − 9.88e6·17-s + 6.31e5·18-s − 5.07e6·19-s − 2.43e7·20-s − 1.83e7·21-s + 8.29e5·22-s + 4.46e7·23-s − 1.03e7·24-s + 1.10e8·25-s + 1.58e7·26-s + 1.43e7·27-s + 1.46e8·28-s + ⋯ |
| L(s) = 1 | + 0.236·2-s + 0.577·3-s − 0.944·4-s + 1.80·5-s + 0.136·6-s − 1.70·7-s − 0.459·8-s + 0.333·9-s + 0.426·10-s + 0.145·11-s − 0.545·12-s + 1.10·13-s − 0.402·14-s + 1.04·15-s + 0.835·16-s − 1.68·17-s + 0.0787·18-s − 0.469·19-s − 1.70·20-s − 0.982·21-s + 0.0342·22-s + 1.44·23-s − 0.265·24-s + 2.25·25-s + 0.261·26-s + 0.192·27-s + 1.60·28-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 | \( 1 - 10.6T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.26e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 7.56e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 7.75e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.47e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 9.88e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 5.07e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.46e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 8.95e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 4.40e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.52e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.31e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.39e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.26e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.83e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.97e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.49e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.84e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.03e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.76e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.60e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.92e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.01e11T + 7.15e21T^{2} \) |
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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
−9.790324362389914158199638704682, −9.229843429834738017743771727697, −8.696112485531838131021072172633, −6.60857182689548228374724590554, −6.19128254987258743371122786314, −4.91838388619518916977402142883, −3.58223164072139175138986386344, −2.68297506313966791050026815080, −1.40825868741763675729884159703, 0,
1.40825868741763675729884159703, 2.68297506313966791050026815080, 3.58223164072139175138986386344, 4.91838388619518916977402142883, 6.19128254987258743371122786314, 6.60857182689548228374724590554, 8.696112485531838131021072172633, 9.229843429834738017743771727697, 9.790324362389914158199638704682
