| L(s) = 1 | − 5.65i·3-s + 7·5-s + 11·7-s − 23.0·9-s + 3·11-s + 11.3i·13-s − 39.5i·15-s − 17·17-s − 19·19-s − 62.2i·21-s + 2·23-s + 24·25-s + 79.1i·27-s + 39.5i·29-s − 5.65i·31-s + ⋯ |
| L(s) = 1 | − 1.88i·3-s + 1.40·5-s + 1.57·7-s − 2.55·9-s + 0.272·11-s + 0.870i·13-s − 2.63i·15-s − 17-s − 19-s − 2.96i·21-s + 0.0869·23-s + 0.959·25-s + 2.93i·27-s + 1.36i·29-s − 0.182i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.33453 - 1.33453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33453 - 1.33453i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 5.65iT - 9T^{2} \) |
| 5 | \( 1 - 7T + 25T^{2} \) |
| 7 | \( 1 - 11T + 49T^{2} \) |
| 11 | \( 1 - 3T + 121T^{2} \) |
| 13 | \( 1 - 11.3iT - 169T^{2} \) |
| 17 | \( 1 + 17T + 289T^{2} \) |
| 23 | \( 1 - 2T + 529T^{2} \) |
| 29 | \( 1 - 39.5iT - 841T^{2} \) |
| 31 | \( 1 + 5.65iT - 961T^{2} \) |
| 37 | \( 1 - 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 21T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 5.65iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 23T + 3.72e3T^{2} \) |
| 67 | \( 1 + 39.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 39T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
| 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
−12.65769332437284218984564154586, −11.63242211949086091319988497813, −10.82030881812000784121619400354, −9.020690091650634069050796913326, −8.296702361817719836853141309311, −6.99786973484652188153231525181, −6.26116617234065239606643343419, −5.02443761752915923068876677726, −2.15736301476957795394401130853, −1.57412450221512034621316246811,
2.34548562951933833460140836405, 4.23226180913465932621949627187, 5.12416259033016196125625692446, 6.03478965060329443379027960019, 8.256644258512608096815076929355, 9.079893014675088962982560744070, 10.06223156599613309973975192926, 10.76210520633472524788783236008, 11.51755903301418387291056793461, 13.28308581399942488969058272116
