| L(s) = 1 | − 5-s + 4·7-s − 4·13-s + 4·19-s − 23-s + 25-s + 6·29-s + 4·31-s − 4·35-s + 2·37-s + 4·43-s + 9·49-s + 6·53-s − 6·59-s + 8·61-s + 4·65-s + 4·67-s − 12·71-s − 10·73-s − 14·79-s + 6·83-s + 6·89-s − 16·91-s − 4·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.10·13-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.676·35-s + 0.328·37-s + 0.609·43-s + 9/7·49-s + 0.824·53-s − 0.781·59-s + 1.02·61-s + 0.496·65-s + 0.488·67-s − 1.42·71-s − 1.17·73-s − 1.57·79-s + 0.658·83-s + 0.635·89-s − 1.67·91-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.430655481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.430655481\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.90877816713429, −15.29713414221835, −14.73797941049309, −14.35225243281815, −13.87761360695544, −13.20163571735103, −12.38307613977490, −11.87345810449355, −11.63058191894739, −10.94119478450909, −10.28697113670456, −9.802917412995569, −8.975823597734071, −8.401291983821556, −7.842472829869366, −7.405040905358967, −6.804736871618991, −5.821138590668929, −5.216649493620907, −4.576862668910519, −4.220311074014939, −3.091711105594494, −2.455072442950816, −1.543648478005036, −0.6996162330381234,
0.6996162330381234, 1.543648478005036, 2.455072442950816, 3.091711105594494, 4.220311074014939, 4.576862668910519, 5.216649493620907, 5.821138590668929, 6.804736871618991, 7.405040905358967, 7.842472829869366, 8.401291983821556, 8.975823597734071, 9.802917412995569, 10.28697113670456, 10.94119478450909, 11.63058191894739, 11.87345810449355, 12.38307613977490, 13.20163571735103, 13.87761360695544, 14.35225243281815, 14.73797941049309, 15.29713414221835, 15.90877816713429
