L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 7-s − 8-s − 2·9-s + 3·10-s − 2·11-s − 12-s − 2·13-s + 14-s + 3·15-s + 16-s − 2·17-s + 2·18-s + 3·19-s − 3·20-s + 21-s + 2·22-s + 24-s + 4·25-s + 2·26-s + 5·27-s − 28-s − 29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.603·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s + 0.471·18-s + 0.688·19-s − 0.670·20-s + 0.218·21-s + 0.426·22-s + 0.204·24-s + 4/5·25-s + 0.392·26-s + 0.962·27-s − 0.188·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.58242193269923729408516201412, −11.74607235447166513054552337542, −11.06328956516952418887798149764, −9.918669122180678226400915428597, −8.541746455043247715836395438509, −7.66729420791456037620478398480, −6.49430854440886602475667135106, −4.93626176492922885828678159907, −3.11821884286468829748286267688, 0,
3.11821884286468829748286267688, 4.93626176492922885828678159907, 6.49430854440886602475667135106, 7.66729420791456037620478398480, 8.541746455043247715836395438509, 9.918669122180678226400915428597, 11.06328956516952418887798149764, 11.74607235447166513054552337542, 12.58242193269923729408516201412