| L(s) = 1 | − 3·3-s + 4·5-s + 7-s + 6·9-s + 2·11-s − 5·13-s − 12·15-s − 4·19-s − 3·21-s − 23-s + 11·25-s − 9·27-s + 3·29-s − 5·31-s − 6·33-s + 4·35-s − 4·37-s + 15·39-s + 5·41-s − 4·43-s + 24·45-s + 11·47-s + 49-s + 8·55-s + 12·57-s − 12·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 0.377·7-s + 2·9-s + 0.603·11-s − 1.38·13-s − 3.09·15-s − 0.917·19-s − 0.654·21-s − 0.208·23-s + 11/5·25-s − 1.73·27-s + 0.557·29-s − 0.898·31-s − 1.04·33-s + 0.676·35-s − 0.657·37-s + 2.40·39-s + 0.780·41-s − 0.609·43-s + 3.57·45-s + 1.60·47-s + 1/7·49-s + 1.07·55-s + 1.58·57-s − 1.56·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−17.07292990420764, −16.64120614192065, −16.02166920187525, −15.07786025496834, −14.59503262603858, −13.98144492184856, −13.37202020463035, −12.60835495135929, −12.33554686492731, −11.71875675930458, −10.93176727815936, −10.42879310848341, −10.11394622493159, −9.336384859887165, −8.912242334726688, −7.701423960700302, −6.882948956349970, −6.568759482554845, −5.705484014173152, −5.552453376201101, −4.728807484087240, −4.239205671857218, −2.720927689091296, −1.891223337571582, −1.247293666124359, 0,
1.247293666124359, 1.891223337571582, 2.720927689091296, 4.239205671857218, 4.728807484087240, 5.552453376201101, 5.705484014173152, 6.568759482554845, 6.882948956349970, 7.701423960700302, 8.912242334726688, 9.336384859887165, 10.11394622493159, 10.42879310848341, 10.93176727815936, 11.71875675930458, 12.33554686492731, 12.60835495135929, 13.37202020463035, 13.98144492184856, 14.59503262603858, 15.07786025496834, 16.02166920187525, 16.64120614192065, 17.07292990420764
