| L(s) = 1 | + 4-s + 5-s − 2·11-s − 13-s + 16-s − 17-s − 19-s + 20-s + 23-s + 29-s + 41-s − 43-s − 2·44-s + 49-s − 52-s − 2·55-s + 64-s − 65-s − 67-s − 68-s + 71-s − 76-s + 80-s − 85-s + 92-s − 95-s − 103-s + ⋯ |
| L(s) = 1 | + 4-s + 5-s − 2·11-s − 13-s + 16-s − 17-s − 19-s + 20-s + 23-s + 29-s + 41-s − 43-s − 2·44-s + 49-s − 52-s − 2·55-s + 64-s − 65-s − 67-s − 68-s + 71-s − 76-s + 80-s − 85-s + 92-s − 95-s − 103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.074505684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074505684\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 + T )^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.98332821954406819926729736269, −10.49809225540244982376634530742, −9.743262152239776875460018324323, −8.496183220853762622292772169540, −7.51061952797034652984045972716, −6.64264709677162462110086688267, −5.67142322826064861440926168592, −4.78240329404948759354773029217, −2.74916842416376406048180467989, −2.19015086381984588647439184289,
2.19015086381984588647439184289, 2.74916842416376406048180467989, 4.78240329404948759354773029217, 5.67142322826064861440926168592, 6.64264709677162462110086688267, 7.51061952797034652984045972716, 8.496183220853762622292772169540, 9.743262152239776875460018324323, 10.49809225540244982376634530742, 10.98332821954406819926729736269
