| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s − 4·11-s − 16-s − 2·17-s − 4·19-s − 20-s + 4·22-s + 25-s + 2·29-s − 5·32-s + 2·34-s + 10·37-s + 4·38-s + 3·40-s + 10·41-s + 4·43-s + 4·44-s + 8·47-s − 7·49-s − 50-s + 10·53-s − 4·55-s − 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s − 1.20·11-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.371·29-s − 0.883·32-s + 0.342·34-s + 1.64·37-s + 0.648·38-s + 0.474·40-s + 1.56·41-s + 0.609·43-s + 0.603·44-s + 1.16·47-s − 49-s − 0.141·50-s + 1.37·53-s − 0.539·55-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−7.74482502996242578854674275677, −7.04115008666205017767300336308, −6.05689229691242628552203135258, −5.53162433165491737601884137045, −4.53790695666728939256074979209, −4.22278793914800246568022071885, −2.85180203351241453209067702234, −2.20712549728076061101604361734, −1.06775515698045606278151011106, 0,
1.06775515698045606278151011106, 2.20712549728076061101604361734, 2.85180203351241453209067702234, 4.22278793914800246568022071885, 4.53790695666728939256074979209, 5.53162433165491737601884137045, 6.05689229691242628552203135258, 7.04115008666205017767300336308, 7.74482502996242578854674275677
