| L(s) = 1 | − 3·3-s + 5-s − 7-s + 6·9-s − 5·11-s − 3·15-s − 17-s + 6·19-s + 3·21-s − 6·23-s + 25-s − 9·27-s − 9·29-s − 4·31-s + 15·33-s − 35-s − 2·37-s + 4·41-s − 10·43-s + 6·45-s − 47-s + 49-s + 3·51-s + 4·53-s − 5·55-s − 18·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s − 1.50·11-s − 0.774·15-s − 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s − 1.73·27-s − 1.67·29-s − 0.718·31-s + 2.61·33-s − 0.169·35-s − 0.328·37-s + 0.624·41-s − 1.52·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s + 0.420·51-s + 0.549·53-s − 0.674·55-s − 2.38·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| 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
−13.87295423902826, −13.37739144003778, −13.02243284143478, −12.52968059176288, −12.11928026171005, −11.51680674429592, −11.12242837389310, −10.68839658209853, −10.17456804862625, −9.728810935815525, −9.439442213780493, −8.513359163995569, −7.873827204529919, −7.271987717819208, −7.048969935921668, −6.158143371933241, −5.849032191104675, −5.398012254926677, −5.046248531758572, −4.400129984306933, −3.633680509562861, −3.020931523271220, −2.084391402739923, −1.596741104142677, −0.5805540485400794, 0,
0.5805540485400794, 1.596741104142677, 2.084391402739923, 3.020931523271220, 3.633680509562861, 4.400129984306933, 5.046248531758572, 5.398012254926677, 5.849032191104675, 6.158143371933241, 7.048969935921668, 7.271987717819208, 7.873827204529919, 8.513359163995569, 9.439442213780493, 9.728810935815525, 10.17456804862625, 10.68839658209853, 11.12242837389310, 11.51680674429592, 12.11928026171005, 12.52968059176288, 13.02243284143478, 13.37739144003778, 13.87295423902826
