| L(s) = 1 | − 9·3-s − 68·5-s + 81·9-s − 388·11-s + 316·13-s + 612·15-s + 1.05e3·17-s − 1.05e3·19-s + 624·23-s + 1.49e3·25-s − 729·27-s + 7.25e3·29-s − 2.29e3·31-s + 3.49e3·33-s + 1.24e4·37-s − 2.84e3·39-s − 5.37e3·41-s + 1.41e4·43-s − 5.50e3·45-s − 4.71e3·47-s − 9.50e3·51-s + 3.78e3·53-s + 2.63e4·55-s + 9.46e3·57-s − 2.52e4·59-s − 2.06e4·61-s − 2.14e4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.21·5-s + 1/3·9-s − 0.966·11-s + 0.518·13-s + 0.702·15-s + 0.886·17-s − 0.668·19-s + 0.245·23-s + 0.479·25-s − 0.192·27-s + 1.60·29-s − 0.429·31-s + 0.558·33-s + 1.49·37-s − 0.299·39-s − 0.499·41-s + 1.16·43-s − 0.405·45-s − 0.311·47-s − 0.511·51-s + 0.184·53-s + 1.17·55-s + 0.385·57-s − 0.944·59-s − 0.711·61-s − 0.630·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 68 T + p^{5} T^{2} \) |
| 11 | \( 1 + 388 T + p^{5} T^{2} \) |
| 13 | \( 1 - 316 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1056 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1052 T + p^{5} T^{2} \) |
| 23 | \( 1 - 624 T + p^{5} T^{2} \) |
| 29 | \( 1 - 250 p T + p^{5} T^{2} \) |
| 31 | \( 1 + 2296 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12426 T + p^{5} T^{2} \) |
| 41 | \( 1 + 5376 T + p^{5} T^{2} \) |
| 43 | \( 1 - 14164 T + p^{5} T^{2} \) |
| 47 | \( 1 + 4712 T + p^{5} T^{2} \) |
| 53 | \( 1 - 3782 T + p^{5} T^{2} \) |
| 59 | \( 1 + 25244 T + p^{5} T^{2} \) |
| 61 | \( 1 + 20668 T + p^{5} T^{2} \) |
| 67 | \( 1 - 49012 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4760 T + p^{5} T^{2} \) |
| 73 | \( 1 - 65264 T + p^{5} T^{2} \) |
| 79 | \( 1 + 49736 T + p^{5} T^{2} \) |
| 83 | \( 1 + 7788 T + p^{5} T^{2} \) |
| 89 | \( 1 + 36904 T + p^{5} T^{2} \) |
| 97 | \( 1 - 98264 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.579995150175436473565911838828, −8.280920248418602921874737033106, −7.83770300406472300144127344855, −6.80854968632410975348193474195, −5.78321255851835710278346628820, −4.75721686232100214965746363280, −3.87430414602856007582115479115, −2.73611113748570120100016771361, −1.03333346673946836957456104025, 0,
1.03333346673946836957456104025, 2.73611113748570120100016771361, 3.87430414602856007582115479115, 4.75721686232100214965746363280, 5.78321255851835710278346628820, 6.80854968632410975348193474195, 7.83770300406472300144127344855, 8.280920248418602921874737033106, 9.579995150175436473565911838828
