L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s + 4·7-s − 3·8-s + 9-s − 10-s − 4·11-s + 12-s − 13-s + 4·14-s + 15-s − 16-s − 6·17-s + 18-s + 4·19-s + 20-s − 4·21-s − 4·22-s + 3·24-s + 25-s − 26-s − 27-s − 4·28-s − 2·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.352579505\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352579505\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.082261118972883527781052820281, −7.36897666424329032701875252090, −6.60210778624633759789110810303, −5.38970556608445428101182889347, −5.29348633024876018571781039514, −4.55136953927478932116484484552, −3.99380668943318620927064680304, −2.88685298415818296975067585884, −1.92689071709246992621950307107, −0.55663933059618251839955284381,
0.55663933059618251839955284381, 1.92689071709246992621950307107, 2.88685298415818296975067585884, 3.99380668943318620927064680304, 4.55136953927478932116484484552, 5.29348633024876018571781039514, 5.38970556608445428101182889347, 6.60210778624633759789110810303, 7.36897666424329032701875252090, 8.082261118972883527781052820281