L(s) = 1 | + (1.31 + 0.759i)2-s + (0.175 + 0.653i)3-s + (0.152 + 0.263i)4-s + (−2.15 + 0.600i)5-s + (−0.265 + 0.991i)6-s + (−1.29 − 2.24i)7-s − 2.57i·8-s + (2.20 − 1.27i)9-s + (−3.28 − 0.845i)10-s + (1.29 + 4.82i)11-s + (−0.145 + 0.145i)12-s + (−2.37 + 2.71i)13-s − 3.93i·14-s + (−0.769 − 1.30i)15-s + (2.25 − 3.91i)16-s + (0.0790 + 0.0211i)17-s + ⋯ |
L(s) = 1 | + (0.929 + 0.536i)2-s + (0.101 + 0.377i)3-s + (0.0761 + 0.131i)4-s + (−0.963 + 0.268i)5-s + (−0.108 + 0.404i)6-s + (−0.490 − 0.849i)7-s − 0.910i·8-s + (0.733 − 0.423i)9-s + (−1.03 − 0.267i)10-s + (0.390 + 1.45i)11-s + (−0.0420 + 0.0420i)12-s + (−0.658 + 0.752i)13-s − 1.05i·14-s + (−0.198 − 0.336i)15-s + (0.564 − 0.977i)16-s + (0.0191 + 0.00513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13022 + 0.384216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13022 + 0.384216i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.15 - 0.600i)T \) |
| 13 | \( 1 + (2.37 - 2.71i)T \) |
good | 2 | \( 1 + (-1.31 - 0.759i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.175 - 0.653i)T + (-2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (1.29 + 2.24i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.29 - 4.82i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.0790 - 0.0211i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (2.71 + 0.726i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (3.91 - 1.05i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-4.31 - 2.49i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.32 + 2.32i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.285 - 0.494i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.0 + 2.69i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.0354 + 0.132i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + 2.30T + 47T^{2} \) |
| 53 | \( 1 + (-6.70 + 6.70i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.694 - 2.59i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.74 + 4.74i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (13.6 + 7.89i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.98 - 7.42i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 - 6.61iT - 73T^{2} \) |
| 79 | \( 1 - 5.71iT - 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 + (17.2 - 4.63i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-4.65 + 2.68i)T + (48.5 - 84.0i)T^{2} \) |
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
−14.95879517144155210948356296317, −14.25497826894405155403355670461, −12.85578651659278866718492321747, −12.09380489874650016083662645168, −10.34763599730206012885872099020, −9.490138731491951046442117701158, −7.30380421153116645098024302015, −6.72524566811793457796565193959, −4.49956729883076661572392720432, −3.95745940483377041925926730570,
2.90650553003474953498185446163, 4.34366041974543086089987128647, 5.89476724545944050483016158509, 7.79476726228576268921747297896, 8.754066786401803656834500774982, 10.67505587459458737370221831995, 11.92392811090760477542103179422, 12.51577756443096479804271676125, 13.39983061301076602649126441707, 14.57495514542939837289480593103