| L(s) = 1 | − 2·5-s − 7-s + 2·13-s − 2·17-s + 4·19-s − 25-s − 6·29-s + 2·35-s + 6·37-s + 6·41-s + 8·43-s − 8·47-s + 49-s − 6·53-s + 12·59-s + 10·61-s − 4·65-s + 16·67-s + 8·71-s − 6·73-s + 8·79-s + 12·83-s + 4·85-s + 14·89-s − 2·91-s − 8·95-s − 6·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 1.11·29-s + 0.338·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.95·67-s + 0.949·71-s − 0.702·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s + 1.48·89-s − 0.209·91-s − 0.820·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286339756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286339756\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.247741063863962618328153365055, −8.217169499557272560754315515660, −7.69091673996266076916622062025, −6.86386789390874373790057825302, −6.01897990892543982446131502516, −5.11532186080382492977915513067, −4.02917005098085719577078580561, −3.49509131771825868682242453713, −2.29213508634424245087777326070, −0.74897918123083417159453737718,
0.74897918123083417159453737718, 2.29213508634424245087777326070, 3.49509131771825868682242453713, 4.02917005098085719577078580561, 5.11532186080382492977915513067, 6.01897990892543982446131502516, 6.86386789390874373790057825302, 7.69091673996266076916622062025, 8.217169499557272560754315515660, 9.247741063863962618328153365055
