| L(s) = 1 | − 3.11·3-s + 0.354·5-s − 0.325·7-s + 6.71·9-s − 5.04·11-s + 6.82·13-s − 1.10·15-s − 3.04·17-s + 1.01·21-s + 1.31·23-s − 4.87·25-s − 11.5·27-s − 2.78·29-s + 7.53·31-s + 15.7·33-s − 0.115·35-s − 5.42·37-s − 21.2·39-s + 7.90·41-s − 7.42·43-s + 2.38·45-s + 7.97·47-s − 6.89·49-s + 9.47·51-s − 9.11·53-s − 1.78·55-s + 7.72·59-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 0.158·5-s − 0.122·7-s + 2.23·9-s − 1.52·11-s + 1.89·13-s − 0.285·15-s − 0.737·17-s + 0.221·21-s + 0.275·23-s − 0.974·25-s − 2.23·27-s − 0.517·29-s + 1.35·31-s + 2.73·33-s − 0.0195·35-s − 0.891·37-s − 3.40·39-s + 1.23·41-s − 1.13·43-s + 0.355·45-s + 1.16·47-s − 0.984·49-s + 1.32·51-s − 1.25·53-s − 0.241·55-s + 1.00·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 - 0.354T + 5T^{2} \) |
| 7 | \( 1 + 0.325T + 7T^{2} \) |
| 11 | \( 1 + 5.04T + 11T^{2} \) |
| 13 | \( 1 - 6.82T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 2.78T + 29T^{2} \) |
| 31 | \( 1 - 7.53T + 31T^{2} \) |
| 37 | \( 1 + 5.42T + 37T^{2} \) |
| 41 | \( 1 - 7.90T + 41T^{2} \) |
| 43 | \( 1 + 7.42T + 43T^{2} \) |
| 47 | \( 1 - 7.97T + 47T^{2} \) |
| 53 | \( 1 + 9.11T + 53T^{2} \) |
| 59 | \( 1 - 7.72T + 59T^{2} \) |
| 61 | \( 1 + 0.534T + 61T^{2} \) |
| 67 | \( 1 - 6.27T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 5.67T + 73T^{2} \) |
| 79 | \( 1 - 9.78T + 79T^{2} \) |
| 83 | \( 1 - 6.93T + 83T^{2} \) |
| 89 | \( 1 + 3.44T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.68013502543191930491239587264, −6.74375968900563876755167046225, −6.21692782648283571888220152238, −5.68745329974646165313726166483, −5.04328276857658674957567320742, −4.31822419602892089223140369646, −3.40727584537276636751772749546, −2.11012525911360454613934082317, −1.03902838075451982530284370331, 0,
1.03902838075451982530284370331, 2.11012525911360454613934082317, 3.40727584537276636751772749546, 4.31822419602892089223140369646, 5.04328276857658674957567320742, 5.68745329974646165313726166483, 6.21692782648283571888220152238, 6.74375968900563876755167046225, 7.68013502543191930491239587264
