| L(s) = 1 | − 3·3-s + 5-s + 7-s + 6·9-s + 5·11-s − 3·13-s − 3·15-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 + 9·39-s + 4·41-s + 10·43-s + 6·45-s − 47-s + 49-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 − 0.832·13-s − 0.774·15-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 + 1.44·39-s + 0.624·41-s + 1.52·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s + 0.549·53-s + 0.674·55-s − 2.38·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.886334929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886334929\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 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 | \( 1 - 13 T + p T^{2} \) |
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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
−14.57760374038232, −14.17616991677408, −13.94825057806990, −13.03462637151245, −12.42573839608324, −12.02109731550054, −11.68685433735700, −11.36334597243416, −10.51999632438927, −10.12168988983702, −9.708843227534562, −9.112697511941738, −8.385600793016169, −7.513525916227198, −7.147269369827093, −6.433201153715929, −6.091382459548121, −5.552508322143550, −4.890507972777944, −4.452555062175684, −3.860797200762069, −2.833639852790370, −1.923986532316487, −1.130155630264092, −0.6727201046413670,
0.6727201046413670, 1.130155630264092, 1.923986532316487, 2.833639852790370, 3.860797200762069, 4.452555062175684, 4.890507972777944, 5.552508322143550, 6.091382459548121, 6.433201153715929, 7.147269369827093, 7.513525916227198, 8.385600793016169, 9.112697511941738, 9.708843227534562, 10.12168988983702, 10.51999632438927, 11.36334597243416, 11.68685433735700, 12.02109731550054, 12.42573839608324, 13.03462637151245, 13.94825057806990, 14.17616991677408, 14.57760374038232
