| L(s) = 1 | + 2-s − 9·3-s − 31·4-s − 34·5-s − 9·6-s − 49·7-s − 63·8-s + 81·9-s − 34·10-s − 340·11-s + 279·12-s + 454·13-s − 49·14-s + 306·15-s + 929·16-s − 798·17-s + 81·18-s + 892·19-s + 1.05e3·20-s + 441·21-s − 340·22-s − 3.19e3·23-s + 567·24-s − 1.96e3·25-s + 454·26-s − 729·27-s + 1.51e3·28-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 0.577·3-s − 0.968·4-s − 0.608·5-s − 0.102·6-s − 0.377·7-s − 0.348·8-s + 1/3·9-s − 0.107·10-s − 0.847·11-s + 0.559·12-s + 0.745·13-s − 0.0668·14-s + 0.351·15-s + 0.907·16-s − 0.669·17-s + 0.0589·18-s + 0.566·19-s + 0.589·20-s + 0.218·21-s − 0.149·22-s − 1.25·23-s + 0.200·24-s − 0.630·25-s + 0.131·26-s − 0.192·27-s + 0.366·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - T + p^{5} T^{2} \) |
| 5 | \( 1 + 34 T + p^{5} T^{2} \) |
| 11 | \( 1 + 340 T + p^{5} T^{2} \) |
| 13 | \( 1 - 454 T + p^{5} T^{2} \) |
| 17 | \( 1 + 798 T + p^{5} T^{2} \) |
| 19 | \( 1 - 892 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3192 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8242 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2496 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9798 T + p^{5} T^{2} \) |
| 41 | \( 1 - 19834 T + p^{5} T^{2} \) |
| 43 | \( 1 + 17236 T + p^{5} T^{2} \) |
| 47 | \( 1 - 8928 T + p^{5} T^{2} \) |
| 53 | \( 1 - 150 T + p^{5} T^{2} \) |
| 59 | \( 1 + 42396 T + p^{5} T^{2} \) |
| 61 | \( 1 - 14758 T + p^{5} T^{2} \) |
| 67 | \( 1 + 1676 T + p^{5} T^{2} \) |
| 71 | \( 1 - 14568 T + p^{5} T^{2} \) |
| 73 | \( 1 - 78378 T + p^{5} T^{2} \) |
| 79 | \( 1 + 2272 T + p^{5} T^{2} \) |
| 83 | \( 1 + 37764 T + p^{5} T^{2} \) |
| 89 | \( 1 + 117286 T + p^{5} T^{2} \) |
| 97 | \( 1 - 10002 T + p^{5} T^{2} \) |
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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
−16.42838682295360959062966332225, −15.33216633212325923755708322977, −13.64937104902993248859833464059, −12.63565727830355611493095022722, −11.15076671357130659061177396747, −9.559142676085913900520360107007, −7.88096433507024607788982289933, −5.73242448063279756033617294087, −3.99730846216034146478381460190, 0,
3.99730846216034146478381460190, 5.73242448063279756033617294087, 7.88096433507024607788982289933, 9.559142676085913900520360107007, 11.15076671357130659061177396747, 12.63565727830355611493095022722, 13.64937104902993248859833464059, 15.33216633212325923755708322977, 16.42838682295360959062966332225
