| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 2·13-s − 14-s − 15-s + 16-s − 6·17-s − 18-s + 8·19-s − 20-s + 21-s − 22-s − 24-s + 25-s − 2·26-s + 27-s + 28-s − 29-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 − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.186079075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.186079075\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.97852136795638, −13.84824197276833, −13.31875933391042, −12.65690630054680, −11.95337458432700, −11.59661423374704, −11.25724261416394, −10.55473315355686, −10.11656731324485, −9.430091139120685, −9.091495687363455, −8.535529540041680, −8.060033914207410, −7.679146751297197, −6.896206945361430, −6.703888648303549, −5.882742450019108, −5.183466176033950, −4.503752460102161, −3.992158893932540, −3.182551691474799, −2.811067030458077, −1.904784380392060, −1.324965435337770, −0.5651496821981244,
0.5651496821981244, 1.324965435337770, 1.904784380392060, 2.811067030458077, 3.182551691474799, 3.992158893932540, 4.503752460102161, 5.183466176033950, 5.882742450019108, 6.703888648303549, 6.896206945361430, 7.679146751297197, 8.060033914207410, 8.535529540041680, 9.091495687363455, 9.430091139120685, 10.11656731324485, 10.55473315355686, 11.25724261416394, 11.59661423374704, 11.95337458432700, 12.65690630054680, 13.31875933391042, 13.84824197276833, 13.97852136795638
