| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 7-s − 3·8-s + 9-s − 10-s + 11-s − 12-s − 6·13-s + 14-s − 15-s − 16-s + 18-s + 8·19-s + 20-s + 21-s + 22-s − 3·24-s + 25-s − 6·26-s + 27-s − 28-s + 10·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + 1.85·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.702074380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.702074380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.55569426003432, −12.26165283390361, −11.92674587065684, −11.33691605495427, −10.90347068187476, −10.19840090734444, −9.667070756715058, −9.421319436599278, −9.082685155091848, −8.345732312937854, −8.018752035455696, −7.384196872151080, −7.296560223680320, −6.456627824838668, −6.016719360960681, −5.193586637861126, −4.975093531213349, −4.639495479753336, −4.010788014760150, −3.433625901482053, −3.083336924141391, −2.537684705136265, −1.907740893175386, −1.011927385377067, −0.4959069286637198,
0.4959069286637198, 1.011927385377067, 1.907740893175386, 2.537684705136265, 3.083336924141391, 3.433625901482053, 4.010788014760150, 4.639495479753336, 4.975093531213349, 5.193586637861126, 6.016719360960681, 6.456627824838668, 7.296560223680320, 7.384196872151080, 8.018752035455696, 8.345732312937854, 9.082685155091848, 9.421319436599278, 9.667070756715058, 10.19840090734444, 10.90347068187476, 11.33691605495427, 11.92674587065684, 12.26165283390361, 12.55569426003432
