| L(s) = 1 | + (0.365 + 1.36i)2-s + (−0.140 − 1.72i)3-s + (−1.73 + 0.998i)4-s + (−0.152 − 1.15i)5-s + (2.30 − 0.822i)6-s + (−0.496 − 1.85i)7-s + (−1.99 − 2.00i)8-s + (−2.96 + 0.484i)9-s + (1.52 − 0.630i)10-s + (3.87 − 2.97i)11-s + (1.96 + 2.85i)12-s + (1.63 − 2.12i)13-s + (2.35 − 1.35i)14-s + (−1.97 + 0.424i)15-s + (2.00 − 3.46i)16-s − 4.48·17-s + ⋯ |
| L(s) = 1 | + (0.258 + 0.966i)2-s + (−0.0810 − 0.996i)3-s + (−0.866 + 0.499i)4-s + (−0.0680 − 0.516i)5-s + (0.941 − 0.335i)6-s + (−0.187 − 0.700i)7-s + (−0.706 − 0.708i)8-s + (−0.986 + 0.161i)9-s + (0.481 − 0.199i)10-s + (1.16 − 0.896i)11-s + (0.567 + 0.823i)12-s + (0.452 − 0.589i)13-s + (0.628 − 0.362i)14-s + (−0.509 + 0.109i)15-s + (0.501 − 0.865i)16-s − 1.08·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04697 - 0.483482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04697 - 0.483482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.365 - 1.36i)T \) |
| 3 | \( 1 + (0.140 + 1.72i)T \) |
| good | 5 | \( 1 + (0.152 + 1.15i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.496 + 1.85i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.87 + 2.97i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.63 + 2.12i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 4.48T + 17T^{2} \) |
| 19 | \( 1 + (-1.00 + 0.415i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (0.329 + 0.0881i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (5.80 + 0.764i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-4.83 + 2.79i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.90 - 7.01i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.987 - 3.68i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-7.23 - 9.43i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (3.25 + 1.88i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.91 - 9.45i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-0.952 + 0.125i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.55 + 11.8i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-1.54 + 2.00i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (6.68 + 6.68i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.787 - 0.787i)T - 73iT^{2} \) |
| 79 | \( 1 + (-3.18 + 5.51i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-17.9 - 2.35i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (2.96 - 2.96i)T - 89iT^{2} \) |
| 97 | \( 1 + (1.67 - 2.90i)T + (-48.5 - 84.0i)T^{2} \) |
| 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
−11.91853609244491439641824830559, −10.97907240851365393359431898880, −9.338018828329871463508664340337, −8.537324827336968124529417159724, −7.73547314478651714709635936523, −6.60774310607320171223530381578, −6.06928852899032181802113598270, −4.66476246318419785328514679160, −3.35750097825787687788021884571, −0.860315797013849820596261123480,
2.20776501262576239110041971145, 3.59626129649268858859205083494, 4.43758698074575954732957471274, 5.65283004111591257379768114427, 6.80859660359334829769506404920, 8.878211107315435312936237511306, 9.132193670975513797125221414258, 10.19781793407511925275622074657, 11.06833999959076291749591009714, 11.74123737539109651637878233152
