| L(s) = 1 | − 5-s + 7-s − 3·9-s − 11-s + 2·13-s + 2·17-s + 25-s + 6·29-s − 35-s − 2·37-s − 6·41-s + 4·43-s + 3·45-s + 49-s − 10·53-s + 55-s − 6·61-s − 3·63-s − 2·65-s − 12·67-s + 8·71-s − 14·73-s − 77-s + 16·79-s + 9·81-s + 16·83-s − 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 1/5·25-s + 1.11·29-s − 0.169·35-s − 0.328·37-s − 0.937·41-s + 0.609·43-s + 0.447·45-s + 1/7·49-s − 1.37·53-s + 0.134·55-s − 0.768·61-s − 0.377·63-s − 0.248·65-s − 1.46·67-s + 0.949·71-s − 1.63·73-s − 0.113·77-s + 1.80·79-s + 81-s + 1.75·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.544488985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.544488985\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.019507912007660627155216613270, −7.57238588025961469976827659711, −6.54552484262536705966493618713, −5.97151542383125234045272239911, −5.14008165083581439444940203982, −4.53157863178380980308572455722, −3.47479146923965024581165259890, −2.94333054874878065580725825412, −1.83673329501046703386845891052, −0.64629607415633286006877787385,
0.64629607415633286006877787385, 1.83673329501046703386845891052, 2.94333054874878065580725825412, 3.47479146923965024581165259890, 4.53157863178380980308572455722, 5.14008165083581439444940203982, 5.97151542383125234045272239911, 6.54552484262536705966493618713, 7.57238588025961469976827659711, 8.019507912007660627155216613270
