| L(s) = 1 | + 0.206·5-s − 3.03i·7-s + 5.02i·11-s − 4.03i·13-s + 4.81i·17-s + 1.03·19-s − 5.43·23-s − 4.95·25-s + 2.90·29-s + 1.23i·31-s − 0.629i·35-s − 8.77i·37-s − 0.979i·41-s − 3.18·43-s + 3.26·47-s + ⋯ |
| L(s) = 1 | + 0.0925·5-s − 1.14i·7-s + 1.51i·11-s − 1.12i·13-s + 1.16i·17-s + 0.238·19-s − 1.13·23-s − 0.991·25-s + 0.539·29-s + 0.222i·31-s − 0.106i·35-s − 1.44i·37-s − 0.152i·41-s − 0.485·43-s + 0.476·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4477256518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4477256518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.206T + 5T^{2} \) |
| 7 | \( 1 + 3.03iT - 7T^{2} \) |
| 11 | \( 1 - 5.02iT - 11T^{2} \) |
| 13 | \( 1 + 4.03iT - 13T^{2} \) |
| 17 | \( 1 - 4.81iT - 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 + 5.43T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 - 1.23iT - 31T^{2} \) |
| 37 | \( 1 + 8.77iT - 37T^{2} \) |
| 41 | \( 1 + 0.979iT - 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 + 7.91T + 53T^{2} \) |
| 59 | \( 1 + 14.8iT - 59T^{2} \) |
| 61 | \( 1 + 9.53iT - 61T^{2} \) |
| 67 | \( 1 + 0.735T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 14.1iT - 79T^{2} \) |
| 83 | \( 1 - 4.23iT - 83T^{2} \) |
| 89 | \( 1 + 11.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
| 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
−7.81391466001295202650114169901, −7.29834423594331662184634267598, −6.49547739236126812043792297286, −5.74167980701007810124093319323, −4.87090325044878557953651929714, −4.07476112168712513956283955664, −3.55836362149768244996129212627, −2.27424355668579559008219914028, −1.46258823735810079918658467031, −0.11510459037814814109699483021,
1.36460973261532128114981639196, 2.46309322424865117219066442419, 3.08796939180106207719549313488, 4.10979428753216451429770576090, 4.94922007134555302647162596631, 5.88338465237339881792086100705, 6.10608281138440001777395330986, 7.07958516405793274228940690520, 7.956186224653072802657638533976, 8.617100888325524845461704388437
