| L(s) = 1 | − 5-s − 6·13-s + 7·17-s − 4·19-s + 3·23-s + 25-s + 4·29-s + 3·31-s − 2·41-s + 4·43-s − 7·47-s − 7·49-s − 5·53-s + 4·59-s + 61-s + 6·65-s + 2·67-s − 10·71-s − 4·73-s − 5·79-s + 12·83-s − 7·85-s + 4·95-s + 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.66·13-s + 1.69·17-s − 0.917·19-s + 0.625·23-s + 1/5·25-s + 0.742·29-s + 0.538·31-s − 0.312·41-s + 0.609·43-s − 1.02·47-s − 49-s − 0.686·53-s + 0.520·59-s + 0.128·61-s + 0.744·65-s + 0.244·67-s − 1.18·71-s − 0.468·73-s − 0.562·79-s + 1.31·83-s − 0.759·85-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.99394941335035, −15.05361787752401, −14.71516882068276, −14.45628069684539, −13.71214847531522, −12.89832844677016, −12.59485075209051, −11.87012339229850, −11.75966388300822, −10.74325795419165, −10.36046640973437, −9.684614519914791, −9.350357891784543, −8.302802694714082, −8.093284824338458, −7.340576209232289, −6.890519600184979, −6.154539359117515, −5.380870934339636, −4.798633689330071, −4.310287567750995, −3.302548396092713, −2.878999056971522, −1.983700585765620, −0.9957648712832646, 0,
0.9957648712832646, 1.983700585765620, 2.878999056971522, 3.302548396092713, 4.310287567750995, 4.798633689330071, 5.380870934339636, 6.154539359117515, 6.890519600184979, 7.340576209232289, 8.093284824338458, 8.302802694714082, 9.350357891784543, 9.684614519914791, 10.36046640973437, 10.74325795419165, 11.75966388300822, 11.87012339229850, 12.59485075209051, 12.89832844677016, 13.71214847531522, 14.45628069684539, 14.71516882068276, 15.05361787752401, 15.99394941335035
