| L(s) = 1 | − 13·2-s + 27·3-s + 105·4-s − 351·6-s + 338·7-s − 533·8-s + 729·9-s + 806·11-s + 2.83e3·12-s + 3.00e3·13-s − 4.39e3·14-s + 209·16-s − 9.77e3·17-s − 9.47e3·18-s + 9.12e3·21-s − 1.04e4·22-s − 1.43e4·24-s + 1.56e4·25-s − 3.90e4·26-s + 1.96e4·27-s + 3.54e4·28-s − 2.43e4·29-s + 3.13e4·32-s + 2.17e4·33-s + 1.27e5·34-s + 7.65e4·36-s + 8.10e4·39-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 3-s + 1.64·4-s − 1.62·6-s + 0.985·7-s − 1.04·8-s + 9-s + 0.605·11-s + 1.64·12-s + 1.36·13-s − 1.60·14-s + 0.0510·16-s − 1.99·17-s − 1.62·18-s + 0.985·21-s − 0.984·22-s − 1.04·24-s + 25-s − 2.22·26-s + 27-s + 1.61·28-s − 29-s + 0.958·32-s + 0.605·33-s + 3.23·34-s + 1.64·36-s + 1.36·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.505981475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505981475\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{3} T \) |
| 29 | \( 1 + p^{3} T \) |
| good | 2 | \( 1 + 13 T + p^{6} T^{2} \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( 1 - 338 T + p^{6} T^{2} \) |
| 11 | \( 1 - 806 T + p^{6} T^{2} \) |
| 13 | \( 1 - 3002 T + p^{6} T^{2} \) |
| 17 | \( 1 + 9778 T + p^{6} T^{2} \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( 1 - 132158 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 - 151502 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( 1 - 267098 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( 1 + 71266 T + p^{6} T^{2} \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−13.09140573170508435263190773643, −11.29684872532188667005732422937, −10.70520428551998704634204931155, −9.129005769015911674123792162491, −8.783756335227331507673381749268, −7.75458868454525756393668949690, −6.62939683996846317625462748322, −4.21039739642878302246659081305, −2.21792129386405074716958977347, −1.12414168368365830004342972037,
1.12414168368365830004342972037, 2.21792129386405074716958977347, 4.21039739642878302246659081305, 6.62939683996846317625462748322, 7.75458868454525756393668949690, 8.783756335227331507673381749268, 9.129005769015911674123792162491, 10.70520428551998704634204931155, 11.29684872532188667005732422937, 13.09140573170508435263190773643
