| L(s) = 1 | − 5-s + 4·7-s − 2·13-s + 6·17-s − 4·19-s + 8·23-s + 25-s − 10·29-s − 4·35-s − 10·37-s − 6·41-s + 8·43-s + 9·49-s + 2·53-s − 12·59-s + 2·61-s + 2·65-s − 12·67-s − 8·71-s + 2·73-s + 8·79-s − 8·83-s − 6·85-s + 6·89-s − 8·91-s + 4·95-s − 6·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.85·29-s − 0.676·35-s − 1.64·37-s − 0.937·41-s + 1.21·43-s + 9/7·49-s + 0.274·53-s − 1.56·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s + 0.234·73-s + 0.900·79-s − 0.878·83-s − 0.650·85-s + 0.635·89-s − 0.838·91-s + 0.410·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.365297660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.365297660\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.77071807451093, −14.30773230072551, −13.75023182074094, −13.12217469381618, −12.37360797072111, −12.24447141520637, −11.45763975302264, −11.11839955854696, −10.59425511534782, −10.13651703205511, −9.234878215208878, −8.873535179295261, −8.288527037444728, −7.629780634176150, −7.405499855141636, −6.814299873208081, −5.782726057801241, −5.405915695884623, −4.783002246877180, −4.346509336012069, −3.498769768530590, −2.984116652928097, −1.936156368541221, −1.532565512006130, −0.5604547439983891,
0.5604547439983891, 1.532565512006130, 1.936156368541221, 2.984116652928097, 3.498769768530590, 4.346509336012069, 4.783002246877180, 5.405915695884623, 5.782726057801241, 6.814299873208081, 7.405499855141636, 7.629780634176150, 8.288527037444728, 8.873535179295261, 9.234878215208878, 10.13651703205511, 10.59425511534782, 11.11839955854696, 11.45763975302264, 12.24447141520637, 12.37360797072111, 13.12217469381618, 13.75023182074094, 14.30773230072551, 14.77071807451093
