L(s) = 1 | + (−1.73 + 3i)2-s + (3.5 − 6.06i)3-s + (−2 − 3.46i)4-s + 13.8·5-s + (12.1 + 21i)6-s + (11.2 + 19.5i)7-s − 13.8·8-s + (−11 − 19.0i)9-s + (−23.9 + 41.5i)10-s + (11.2 − 19.5i)11-s − 28.0·12-s − 78·14-s + (48.4 − 84i)15-s + (39.9 − 69.2i)16-s + (13.5 + 23.3i)17-s + 76.2·18-s + ⋯ |
L(s) = 1 | + (−0.612 + 1.06i)2-s + (0.673 − 1.16i)3-s + (−0.250 − 0.433i)4-s + 1.23·5-s + (0.824 + 1.42i)6-s + (0.607 + 1.05i)7-s − 0.612·8-s + (−0.407 − 0.705i)9-s + (−0.758 + 1.31i)10-s + (0.308 − 0.534i)11-s − 0.673·12-s − 1.48·14-s + (0.834 − 1.44i)15-s + (0.624 − 1.08i)16-s + (0.192 + 0.333i)17-s + 0.997·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.84727 + 0.828701i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.84727 + 0.828701i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (1.73 - 3i)T + (-4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-3.5 + 6.06i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 + (-11.2 - 19.5i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-11.2 + 19.5i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-13.5 - 23.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-44.1 - 76.5i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-28.5 + 49.3i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-34.5 + 59.7i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 72.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-19.9 + 34.5i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (196. - 340.5i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (42.5 + 73.6i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 426T + 1.48e5T^{2} \) |
| 59 | \( 1 + (9.52 + 16.5i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-8.5 - 14.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-82.2 + 142.5i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (291. + 505.5i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (153. - 265.5i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (617. + 1.06e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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
−12.60349263982830510044770485056, −11.73669423240321516893975729681, −9.982581888653905024977317922823, −8.852435582945891632986195043265, −8.367453648465941067597285637160, −7.34105417824589730900377208084, −6.21017703840753424181191599403, −5.57835254209185426275708712413, −2.75587944190217775525305487949, −1.53757732922171221992105764468,
1.31248309036038851134004631207, 2.69994388055419531329030054709, 4.03323144945382492288956548147, 5.35242030642084998889721545685, 7.10221872891256214390298377305, 8.783186784655053790692296812578, 9.450984162130675076569323746027, 10.16224787286871200564989302527, 10.70844899207600945582929921921, 11.81228725481819621235632891538