| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s + 4·11-s − 12-s − 6·13-s + 15-s + 16-s − 2·17-s + 18-s − 4·19-s − 20-s + 4·22-s − 23-s − 24-s + 25-s − 6·26-s − 27-s − 2·29-s + 30-s + 8·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 1.66·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s + 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.08505810276954, −14.85160379860438, −14.18791503105807, −13.79084322036926, −13.00438009720836, −12.46435006455585, −12.16964135432697, −11.68931753124963, −11.16146122522723, −10.66571887972789, −9.908303382167280, −9.547844362194358, −8.781150571695424, −8.096276251787640, −7.483074596800507, −6.860786740203729, −6.478275960101772, −5.942179740527934, −4.989662298615714, −4.722462839259859, −4.082213264564421, −3.529898283291092, −2.546942425513973, −2.022379393092562, −0.9858007129681497, 0,
0.9858007129681497, 2.022379393092562, 2.546942425513973, 3.529898283291092, 4.082213264564421, 4.722462839259859, 4.989662298615714, 5.942179740527934, 6.478275960101772, 6.860786740203729, 7.483074596800507, 8.096276251787640, 8.781150571695424, 9.547844362194358, 9.908303382167280, 10.66571887972789, 11.16146122522723, 11.68931753124963, 12.16964135432697, 12.46435006455585, 13.00438009720836, 13.79084322036926, 14.18791503105807, 14.85160379860438, 15.08505810276954
