L(s) = 1 | + 6·2-s − 92·4-s + 64·7-s − 1.32e3·8-s + 948·11-s + 5.09e3·13-s + 384·14-s + 3.85e3·16-s + 2.83e4·17-s − 8.62e3·19-s + 5.68e3·22-s − 1.52e4·23-s + 3.05e4·26-s − 5.88e3·28-s − 3.65e4·29-s − 2.76e5·31-s + 1.92e5·32-s + 1.70e5·34-s − 2.68e5·37-s − 5.17e4·38-s + 6.29e5·41-s − 6.85e5·43-s − 8.72e4·44-s − 9.17e4·46-s + 5.83e5·47-s − 8.19e5·49-s − 4.69e5·52-s + ⋯ |
L(s) = 1 | + 0.530·2-s − 0.718·4-s + 0.0705·7-s − 0.911·8-s + 0.214·11-s + 0.643·13-s + 0.0374·14-s + 0.235·16-s + 1.40·17-s − 0.288·19-s + 0.113·22-s − 0.262·23-s + 0.341·26-s − 0.0506·28-s − 0.277·29-s − 1.66·31-s + 1.03·32-s + 0.743·34-s − 0.871·37-s − 0.152·38-s + 1.42·41-s − 1.31·43-s − 0.154·44-s − 0.138·46-s + 0.819·47-s − 0.995·49-s − 0.462·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 3 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 64 T + p^{7} T^{2} \) |
| 11 | \( 1 - 948 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5098 T + p^{7} T^{2} \) |
| 17 | \( 1 - 28386 T + p^{7} T^{2} \) |
| 19 | \( 1 + 8620 T + p^{7} T^{2} \) |
| 23 | \( 1 + 15288 T + p^{7} T^{2} \) |
| 29 | \( 1 + 36510 T + p^{7} T^{2} \) |
| 31 | \( 1 + 276808 T + p^{7} T^{2} \) |
| 37 | \( 1 + 268526 T + p^{7} T^{2} \) |
| 41 | \( 1 - 629718 T + p^{7} T^{2} \) |
| 43 | \( 1 + 685772 T + p^{7} T^{2} \) |
| 47 | \( 1 - 583296 T + p^{7} T^{2} \) |
| 53 | \( 1 + 428058 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1306380 T + p^{7} T^{2} \) |
| 61 | \( 1 - 300662 T + p^{7} T^{2} \) |
| 67 | \( 1 - 507244 T + p^{7} T^{2} \) |
| 71 | \( 1 + 5560632 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1369082 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6913720 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4376748 T + p^{7} T^{2} \) |
| 89 | \( 1 - 8528310 T + p^{7} T^{2} \) |
| 97 | \( 1 - 8826814 T + p^{7} T^{2} \) |
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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
−10.48220739083952401567796680959, −9.454575366419476483621309686856, −8.605112063627316628366701710250, −7.53118232572431737988300703368, −6.09154458004630527627844170294, −5.28656742260754299662132358906, −4.06602977757791761397196072567, −3.20062292759901484798136305874, −1.42301116654435969034221949571, 0,
1.42301116654435969034221949571, 3.20062292759901484798136305874, 4.06602977757791761397196072567, 5.28656742260754299662132358906, 6.09154458004630527627844170294, 7.53118232572431737988300703368, 8.605112063627316628366701710250, 9.454575366419476483621309686856, 10.48220739083952401567796680959