| L(s) = 1 | + 4·5-s + 2·7-s − 11-s + 17-s − 6·23-s + 11·25-s − 2·29-s − 4·31-s + 8·35-s − 2·37-s + 6·41-s − 4·43-s − 6·47-s − 3·49-s + 8·53-s − 4·55-s − 8·59-s + 8·61-s − 4·67-s − 6·71-s + 10·73-s − 2·77-s + 6·79-s + 4·83-s + 4·85-s + 14·89-s + 14·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s − 0.301·11-s + 0.242·17-s − 1.25·23-s + 11/5·25-s − 0.371·29-s − 0.718·31-s + 1.35·35-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s + 1.09·53-s − 0.539·55-s − 1.04·59-s + 1.02·61-s − 0.488·67-s − 0.712·71-s + 1.17·73-s − 0.227·77-s + 0.675·79-s + 0.439·83-s + 0.433·85-s + 1.48·89-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.167325865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.167325865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.80194114808004, −13.16100664910039, −12.90523106771932, −12.29299965405590, −11.68199701894927, −11.22678271659753, −10.56902576623384, −10.23067383283974, −9.860915920134437, −9.167999941072623, −8.955797155252508, −8.190410323529046, −7.741493669279718, −7.188840619253390, −6.373554690901721, −6.147048241969214, −5.518191460413204, −5.097230684496969, −4.647919467741798, −3.791290487932412, −3.176802342595121, −2.307129551231201, −1.993928435346790, −1.488588792641115, −0.6099517991653520,
0.6099517991653520, 1.488588792641115, 1.993928435346790, 2.307129551231201, 3.176802342595121, 3.791290487932412, 4.647919467741798, 5.097230684496969, 5.518191460413204, 6.147048241969214, 6.373554690901721, 7.188840619253390, 7.741493669279718, 8.190410323529046, 8.955797155252508, 9.167999941072623, 9.860915920134437, 10.23067383283974, 10.56902576623384, 11.22678271659753, 11.68199701894927, 12.29299965405590, 12.90523106771932, 13.16100664910039, 13.80194114808004
