| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 6·11-s − 2·13-s + 14-s + 16-s − 3·17-s + 2·19-s + 6·22-s − 9·23-s + 2·26-s − 28-s − 6·29-s − 4·31-s − 32-s + 3·34-s − 8·37-s − 2·38-s + 9·41-s + 43-s − 6·44-s + 9·46-s + 6·47-s + 49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.80·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 1.27·22-s − 1.87·23-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s − 1.31·37-s − 0.324·38-s + 1.40·41-s + 0.152·43-s − 0.904·44-s + 1.32·46-s + 0.875·47-s + 1/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3069122310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3069122310\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.55204659218807783152452318309, −7.44186478872568937778700610136, −6.34553804696903276169831576180, −5.72676069452141485473719212086, −5.10031866754772636137102598536, −4.16372073379951617009781050559, −3.24549852528143632157582313598, −2.42753850228122293834169310591, −1.84857906412072979582439986445, −0.27393595083893593215612516992,
0.27393595083893593215612516992, 1.84857906412072979582439986445, 2.42753850228122293834169310591, 3.24549852528143632157582313598, 4.16372073379951617009781050559, 5.10031866754772636137102598536, 5.72676069452141485473719212086, 6.34553804696903276169831576180, 7.44186478872568937778700610136, 7.55204659218807783152452318309
