| L(s) = 1 | + 22.0·2-s + 27·3-s + 357.·4-s − 492.·5-s + 594.·6-s − 1.20e3·7-s + 5.04e3·8-s + 729·9-s − 1.08e4·10-s − 7.75e3·11-s + 9.64e3·12-s + 1.92e3·13-s − 2.66e4·14-s − 1.32e4·15-s + 6.55e4·16-s − 2.18e4·17-s + 1.60e4·18-s − 3.16e4·19-s − 1.75e5·20-s − 3.26e4·21-s − 1.70e5·22-s + 7.60e4·23-s + 1.36e5·24-s + 1.64e5·25-s + 4.24e4·26-s + 1.96e4·27-s − 4.32e5·28-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 0.577·3-s + 2.79·4-s − 1.76·5-s + 1.12·6-s − 1.33·7-s + 3.48·8-s + 0.333·9-s − 3.43·10-s − 1.75·11-s + 1.61·12-s + 0.243·13-s − 2.59·14-s − 1.01·15-s + 3.99·16-s − 1.07·17-s + 0.649·18-s − 1.05·19-s − 4.91·20-s − 0.769·21-s − 3.42·22-s + 1.30·23-s + 2.01·24-s + 2.10·25-s + 0.473·26-s + 0.192·27-s − 3.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 22.0T + 128T^{2} \) |
| 5 | \( 1 + 492.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.20e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.75e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.18e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.16e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.60e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.07e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.68e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 7.25e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.98e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.27e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.07e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.52e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.40e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.06e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.89e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.97e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.98e6T + 8.07e13T^{2} \) |
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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
−11.16934667234787029820610000777, −10.49370406107794386530056373973, −8.465145159218053448488990120458, −7.34444371162114590254377024430, −6.69275187285671834482945236232, −5.16435490462047926506568873397, −4.09180821935133240456614653450, −3.31621821805895061660394435972, −2.50497578735723484853619329653, 0,
2.50497578735723484853619329653, 3.31621821805895061660394435972, 4.09180821935133240456614653450, 5.16435490462047926506568873397, 6.69275187285671834482945236232, 7.34444371162114590254377024430, 8.465145159218053448488990120458, 10.49370406107794386530056373973, 11.16934667234787029820610000777
