| L(s) = 1 | − 8·2-s + 66·3-s + 64·4-s − 528·6-s + 343·7-s − 512·8-s + 2.16e3·9-s + 40·11-s + 4.22e3·12-s + 4.45e3·13-s − 2.74e3·14-s + 4.09e3·16-s − 3.65e4·17-s − 1.73e4·18-s − 4.62e4·19-s + 2.26e4·21-s − 320·22-s + 1.05e5·23-s − 3.37e4·24-s − 3.56e4·26-s − 1.18e3·27-s + 2.19e4·28-s − 1.26e5·29-s − 1.70e5·31-s − 3.27e4·32-s + 2.64e3·33-s + 2.92e5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.41·3-s + 1/2·4-s − 0.997·6-s + 0.377·7-s − 0.353·8-s + 0.991·9-s + 0.00906·11-s + 0.705·12-s + 0.562·13-s − 0.267·14-s + 1/4·16-s − 1.80·17-s − 0.701·18-s − 1.54·19-s + 0.533·21-s − 0.00640·22-s + 1.80·23-s − 0.498·24-s − 0.397·26-s − 0.0116·27-s + 0.188·28-s − 0.961·29-s − 1.03·31-s − 0.176·32-s + 0.0127·33-s + 1.27·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 - 22 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 40 T + p^{7} T^{2} \) |
| 13 | \( 1 - 4452 T + p^{7} T^{2} \) |
| 17 | \( 1 + 36502 T + p^{7} T^{2} \) |
| 19 | \( 1 + 46222 T + p^{7} T^{2} \) |
| 23 | \( 1 - 105200 T + p^{7} T^{2} \) |
| 29 | \( 1 + 126334 T + p^{7} T^{2} \) |
| 31 | \( 1 + 170964 T + p^{7} T^{2} \) |
| 37 | \( 1 + 20954 T + p^{7} T^{2} \) |
| 41 | \( 1 - 318486 T + p^{7} T^{2} \) |
| 43 | \( 1 + 1808 p T + p^{7} T^{2} \) |
| 47 | \( 1 + 703716 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1603278 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1171894 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2068872 T + p^{7} T^{2} \) |
| 67 | \( 1 - 994268 T + p^{7} T^{2} \) |
| 71 | \( 1 - 33280 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2971454 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2376168 T + p^{7} T^{2} \) |
| 83 | \( 1 - 2122358 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6920346 T + p^{7} T^{2} \) |
| 97 | \( 1 + 4952710 T + p^{7} T^{2} \) |
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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
−9.331627754983424850030323532723, −8.929329631039072101529272677608, −8.230804294664376760791036343341, −7.27405646556886067698859722421, −6.31795398871141512082239169514, −4.64846499448934456262580936141, −3.50072261056919248303224992517, −2.37794833653125523952622491101, −1.61691062790273732025959577832, 0,
1.61691062790273732025959577832, 2.37794833653125523952622491101, 3.50072261056919248303224992517, 4.64846499448934456262580936141, 6.31795398871141512082239169514, 7.27405646556886067698859722421, 8.230804294664376760791036343341, 8.929329631039072101529272677608, 9.331627754983424850030323532723
