| L(s) = 1 | + 4.77·7-s + 2.15·11-s + 6.09·13-s + 5.27·17-s − 3.27·19-s − 3.27·23-s + 6.09·29-s + 3·31-s + 2.62·37-s − 8.71·41-s + 10.3·43-s + 4.54·47-s + 15.8·49-s − 9.54·53-s − 3.46·59-s + 2.72·61-s − 2.62·67-s + 6.09·71-s − 2.15·73-s + 10.2·77-s − 1.27·79-s − 15.5·83-s − 10.3·89-s + 29.0·91-s − 16.1·97-s − 2.98·101-s − 16.4·103-s + ⋯ |
| L(s) = 1 | + 1.80·7-s + 0.648·11-s + 1.68·13-s + 1.27·17-s − 0.751·19-s − 0.682·23-s + 1.13·29-s + 0.538·31-s + 0.431·37-s − 1.36·41-s + 1.58·43-s + 0.663·47-s + 2.26·49-s − 1.31·53-s − 0.450·59-s + 0.348·61-s − 0.320·67-s + 0.722·71-s − 0.251·73-s + 1.17·77-s − 0.143·79-s − 1.70·83-s − 1.10·89-s + 3.05·91-s − 1.63·97-s − 0.297·101-s − 1.62·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.148127195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.148127195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 8.71T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2.72T + 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 - 6.09T + 71T^{2} \) |
| 73 | \( 1 + 2.15T + 73T^{2} \) |
| 79 | \( 1 + 1.27T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
| 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
−8.333107483423382982917669845621, −7.67631786708520623821618365013, −6.68718091649763184445197402763, −5.96395781570347091367448424927, −5.32789305805261204759040745575, −4.35278035397212281253843900831, −3.94639786941955030932510199368, −2.79596926315248789199287762356, −1.58032710934954147072065298929, −1.13360164366480695479661551022,
1.13360164366480695479661551022, 1.58032710934954147072065298929, 2.79596926315248789199287762356, 3.94639786941955030932510199368, 4.35278035397212281253843900831, 5.32789305805261204759040745575, 5.96395781570347091367448424927, 6.68718091649763184445197402763, 7.67631786708520623821618365013, 8.333107483423382982917669845621
