| L(s) = 1 | + 4·2-s + 16·4-s − 54·5-s + 49·7-s + 64·8-s − 216·10-s − 594·11-s + 26·13-s + 196·14-s + 256·16-s + 534·17-s − 3.00e3·19-s − 864·20-s − 2.37e3·22-s − 3.51e3·23-s − 209·25-s + 104·26-s + 784·28-s − 4.29e3·29-s + 8.03e3·31-s + 1.02e3·32-s + 2.13e3·34-s − 2.64e3·35-s − 502·37-s − 1.20e4·38-s − 3.45e3·40-s − 9.87e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.965·5-s + 0.377·7-s + 0.353·8-s − 0.683·10-s − 1.48·11-s + 0.0426·13-s + 0.267·14-s + 1/4·16-s + 0.448·17-s − 1.90·19-s − 0.482·20-s − 1.04·22-s − 1.38·23-s − 0.0668·25-s + 0.0301·26-s + 0.188·28-s − 0.948·29-s + 1.50·31-s + 0.176·32-s + 0.316·34-s − 0.365·35-s − 0.0602·37-s − 1.34·38-s − 0.341·40-s − 0.916·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 11 | \( 1 + 54 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 534 T + p^{5} T^{2} \) |
| 19 | \( 1 + 3004 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3510 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4296 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8036 T + p^{5} T^{2} \) |
| 37 | \( 1 + 502 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9870 T + p^{5} T^{2} \) |
| 43 | \( 1 - 9068 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1140 T + p^{5} T^{2} \) |
| 53 | \( 1 + 28356 T + p^{5} T^{2} \) |
| 59 | \( 1 - 8196 T + p^{5} T^{2} \) |
| 61 | \( 1 - 29822 T + p^{5} T^{2} \) |
| 67 | \( 1 + 62884 T + p^{5} T^{2} \) |
| 71 | \( 1 - 34398 T + p^{5} T^{2} \) |
| 73 | \( 1 - 56990 T + p^{5} T^{2} \) |
| 79 | \( 1 - 49496 T + p^{5} T^{2} \) |
| 83 | \( 1 - 52512 T + p^{5} T^{2} \) |
| 89 | \( 1 - 48282 T + p^{5} T^{2} \) |
| 97 | \( 1 + 83938 T + p^{5} T^{2} \) |
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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
−12.06102783297192808642463093832, −11.05125161415643692111545595503, −10.18262364597618560451493802223, −8.276994891691953988989721329367, −7.70383461166281154248829046383, −6.20272755459477013727480472800, −4.87782170820255959675788769451, −3.80920351568475072740392367541, −2.27002282334906808251958792217, 0,
2.27002282334906808251958792217, 3.80920351568475072740392367541, 4.87782170820255959675788769451, 6.20272755459477013727480472800, 7.70383461166281154248829046383, 8.276994891691953988989721329367, 10.18262364597618560451493802223, 11.05125161415643692111545595503, 12.06102783297192808642463093832
