| L(s) = 1 | + 143.·2-s − 729·3-s + 1.23e4·4-s − 1.04e5·6-s + 1.15e5·7-s + 6.02e5·8-s + 5.31e5·9-s − 1.68e6·11-s − 9.03e6·12-s − 8.38e6·13-s + 1.66e7·14-s − 1.50e7·16-s − 1.18e8·17-s + 7.62e7·18-s + 2.93e6·19-s − 8.43e7·21-s − 2.42e8·22-s + 1.14e9·23-s − 4.39e8·24-s − 1.20e9·26-s − 3.87e8·27-s + 1.43e9·28-s − 2.58e9·29-s − 5.28e9·31-s − 7.09e9·32-s + 1.23e9·33-s − 1.69e10·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.51·4-s − 0.915·6-s + 0.371·7-s + 0.812·8-s + 0.333·9-s − 0.287·11-s − 0.873·12-s − 0.481·13-s + 0.589·14-s − 0.224·16-s − 1.18·17-s + 0.528·18-s + 0.0143·19-s − 0.214·21-s − 0.455·22-s + 1.61·23-s − 0.469·24-s − 0.763·26-s − 0.192·27-s + 0.562·28-s − 0.805·29-s − 1.06·31-s − 1.16·32-s + 0.165·33-s − 1.88·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 729T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 143.T + 8.19e3T^{2} \) |
| 7 | \( 1 - 1.15e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 1.68e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 8.38e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.18e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.93e6T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.14e9T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.58e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 5.28e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.25e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.08e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.84e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 8.91e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.29e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 1.48e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 8.88e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.15e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.39e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.23e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 4.29e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.77e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.52e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.54e13T + 6.73e25T^{2} \) |
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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
−11.47197899054271955286496913803, −10.88278202204084591031807716736, −9.171813605011118671510444898612, −7.38789908847719584129353494540, −6.34983511894540870452476884823, −5.18622136622760967090048353730, −4.51097970021768362150106124804, −3.14012585488185655566834427668, −1.83250742963324406530175750639, 0,
1.83250742963324406530175750639, 3.14012585488185655566834427668, 4.51097970021768362150106124804, 5.18622136622760967090048353730, 6.34983511894540870452476884823, 7.38789908847719584129353494540, 9.171813605011118671510444898612, 10.88278202204084591031807716736, 11.47197899054271955286496913803
