| L(s) = 1 | + 5-s + 4·11-s − 2·13-s − 2·19-s + 4·23-s + 25-s + 10·29-s + 4·31-s + 2·37-s + 12·41-s + 4·43-s − 4·47-s + 2·53-s + 4·55-s + 10·59-s − 6·61-s − 2·65-s − 4·67-s + 12·71-s − 4·73-s − 4·79-s + 14·83-s − 8·89-s − 2·95-s − 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s − 0.554·13-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 0.328·37-s + 1.87·41-s + 0.609·43-s − 0.583·47-s + 0.274·53-s + 0.539·55-s + 1.30·59-s − 0.768·61-s − 0.248·65-s − 0.488·67-s + 1.42·71-s − 0.468·73-s − 0.450·79-s + 1.53·83-s − 0.847·89-s − 0.205·95-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.046369015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.046369015\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.38303995763218, −12.89669187292664, −12.42401876388279, −12.04549662801691, −11.47280125656907, −11.07317437059789, −10.44228176880312, −10.02901040913129, −9.539548776393940, −9.016180552301278, −8.705241399124739, −8.055325881194570, −7.489608667827174, −6.927517246528918, −6.359512707985212, −6.201586789455317, −5.391302846677051, −4.802062994168276, −4.363398530959575, −3.841279565203706, −2.996624862333998, −2.592122659579697, −1.928849533867471, −1.076905989800122, −0.7007844686777907,
0.7007844686777907, 1.076905989800122, 1.928849533867471, 2.592122659579697, 2.996624862333998, 3.841279565203706, 4.363398530959575, 4.802062994168276, 5.391302846677051, 6.201586789455317, 6.359512707985212, 6.927517246528918, 7.489608667827174, 8.055325881194570, 8.705241399124739, 9.016180552301278, 9.539548776393940, 10.02901040913129, 10.44228176880312, 11.07317437059789, 11.47280125656907, 12.04549662801691, 12.42401876388279, 12.89669187292664, 13.38303995763218
