| L(s) = 1 | − 18.2·2-s − 80.5·3-s + 204.·4-s + 1.46e3·6-s − 343·7-s − 1.38e3·8-s + 4.29e3·9-s − 6.26e3·11-s − 1.64e4·12-s − 4.90e3·13-s + 6.25e3·14-s − 822.·16-s − 3.37e4·17-s − 7.83e4·18-s + 4.11e4·19-s + 2.76e4·21-s + 1.14e5·22-s + 7.86e3·23-s + 1.11e5·24-s + 8.93e4·26-s − 1.70e5·27-s − 7.00e4·28-s − 5.52e4·29-s + 5.96e3·31-s + 1.92e5·32-s + 5.04e5·33-s + 6.14e5·34-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 1.72·3-s + 1.59·4-s + 2.77·6-s − 0.377·7-s − 0.959·8-s + 1.96·9-s − 1.41·11-s − 2.74·12-s − 0.619·13-s + 0.608·14-s − 0.0501·16-s − 1.66·17-s − 3.16·18-s + 1.37·19-s + 0.650·21-s + 2.28·22-s + 0.134·23-s + 1.65·24-s + 0.997·26-s − 1.66·27-s − 0.602·28-s − 0.420·29-s + 0.0359·31-s + 1.03·32-s + 2.44·33-s + 2.68·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 18.2T + 128T^{2} \) |
| 3 | \( 1 + 80.5T + 2.18e3T^{2} \) |
| 11 | \( 1 + 6.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.90e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.37e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.11e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.86e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.52e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.96e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 6.14e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.64e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.60e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.34e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.85e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.68e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 9.36e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.28e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.78e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.36e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.67e6T + 8.07e13T^{2} \) |
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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
−10.79541264782947173068571441514, −10.06010536836611853952349418238, −9.146200352443663941554428710589, −7.65252794918649991411918844489, −6.97279173765458260543251745051, −5.81286721366198387124679830385, −4.71853613760315620980805919237, −2.35250660479193596483008769541, −0.78341659149201767975791179923, 0,
0.78341659149201767975791179923, 2.35250660479193596483008769541, 4.71853613760315620980805919237, 5.81286721366198387124679830385, 6.97279173765458260543251745051, 7.65252794918649991411918844489, 9.146200352443663941554428710589, 10.06010536836611853952349418238, 10.79541264782947173068571441514
