| L(s) = 1 | − 2-s + 4-s + 3.71i·5-s − 7-s − 8-s − 3.71i·10-s + 1.06i·11-s − 4.70i·13-s + 14-s + 16-s + 1.06i·17-s + (−3.96 − 1.82i)19-s + 3.71i·20-s − 1.06i·22-s − 5.31i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.66i·5-s − 0.377·7-s − 0.353·8-s − 1.17i·10-s + 0.322i·11-s − 1.30i·13-s + 0.267·14-s + 0.250·16-s + 0.259i·17-s + (−0.908 − 0.417i)19-s + 0.831i·20-s − 0.227i·22-s − 1.10i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5000581429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5000581429\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + (3.96 + 1.82i)T \) |
| good | 5 | \( 1 - 3.71iT - 5T^{2} \) |
| 11 | \( 1 - 1.06iT - 11T^{2} \) |
| 13 | \( 1 + 4.70iT - 13T^{2} \) |
| 17 | \( 1 - 1.06iT - 17T^{2} \) |
| 23 | \( 1 + 5.31iT - 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 - 2.65iT - 31T^{2} \) |
| 37 | \( 1 + 3.19iT - 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 9.41iT - 47T^{2} \) |
| 53 | \( 1 + 8.68T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 4.10iT - 67T^{2} \) |
| 71 | \( 1 - 5.14T + 71T^{2} \) |
| 73 | \( 1 + 7.00T + 73T^{2} \) |
| 79 | \( 1 + 3.66iT - 79T^{2} \) |
| 83 | \( 1 - 9.63iT - 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 + 13.1iT - 97T^{2} \) |
| 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
−8.552153349180194496983333086890, −8.200175811965234288128020903646, −7.04336117092769262227357332369, −6.74445942770070748272801152279, −6.04564212063893494111308093412, −4.88328420061580708102098975746, −3.51047144917656134861960787787, −2.91142497508636568948886736646, −2.04424728171556886115363530564, −0.22696222444276819012088982547,
1.14453014272105780086333338183, 1.97966436534369413539468654245, 3.42340517050067233388037291553, 4.45134792329094608926509249847, 5.12729988833709364995817427073, 6.18103858752357099908267091890, 6.79685066660953491758082744538, 7.944222749267814417656402624628, 8.481161449107441200747507270961, 9.061831964448173555086968509550
