| L(s) = 1 | + 2-s − 3-s + 4-s − 4·5-s − 6-s + 7-s + 8-s + 9-s − 4·10-s − 11-s − 12-s − 6·13-s + 14-s + 4·15-s + 16-s − 4·17-s + 18-s − 2·19-s − 4·20-s − 21-s − 22-s − 8·23-s − 24-s + 11·25-s − 6·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.458·19-s − 0.894·20-s − 0.218·21-s − 0.213·22-s − 1.66·23-s − 0.204·24-s + 11/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.99555820546878314855420687836, −10.01228279213210772511711884000, −8.515253748126092137293786902708, −7.56853928475057757956492711754, −7.06196128452260537589494409619, −5.68884256231510340821014445655, −4.50182617970418156245344086250, −4.10732223636165202223499092609, −2.48203239562124190952326585155, 0,
2.48203239562124190952326585155, 4.10732223636165202223499092609, 4.50182617970418156245344086250, 5.68884256231510340821014445655, 7.06196128452260537589494409619, 7.56853928475057757956492711754, 8.515253748126092137293786902708, 10.01228279213210772511711884000, 10.99555820546878314855420687836
