| L(s) = 1 | + (1.75 + 1.50i)2-s + (−2.15 + 1.14i)3-s + (0.498 + 3.30i)4-s + (−0.510 + 2.17i)5-s + (−5.50 − 1.25i)6-s + (1.70 − 2.02i)7-s + (−1.65 + 2.63i)8-s + (1.66 − 2.44i)9-s + (−4.17 + 3.04i)10-s + (−1.76 + 1.20i)11-s + (−4.85 − 6.57i)12-s + (−2.38 + 0.833i)13-s + (6.04 − 0.963i)14-s + (−1.38 − 5.28i)15-s + (−0.484 + 0.149i)16-s + (−0.155 + 0.356i)17-s + ⋯ |
| L(s) = 1 | + (1.23 + 1.06i)2-s + (−1.24 + 0.658i)3-s + (0.249 + 1.65i)4-s + (−0.228 + 0.973i)5-s + (−2.24 − 0.512i)6-s + (0.645 − 0.763i)7-s + (−0.585 + 0.932i)8-s + (0.555 − 0.814i)9-s + (−1.32 + 0.962i)10-s + (−0.531 + 0.362i)11-s + (−1.40 − 1.89i)12-s + (−0.660 + 0.231i)13-s + (1.61 − 0.257i)14-s + (−0.356 − 1.36i)15-s + (−0.121 + 0.0373i)16-s + (−0.0377 + 0.0864i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.250753 + 1.51120i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.250753 + 1.51120i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.510 - 2.17i)T \) |
| 7 | \( 1 + (-1.70 + 2.02i)T \) |
| good | 2 | \( 1 + (-1.75 - 1.50i)T + (0.298 + 1.97i)T^{2} \) |
| 3 | \( 1 + (2.15 - 1.14i)T + (1.68 - 2.47i)T^{2} \) |
| 11 | \( 1 + (1.76 - 1.20i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (2.38 - 0.833i)T + (10.1 - 8.10i)T^{2} \) |
| 17 | \( 1 + (0.155 - 0.356i)T + (-11.5 - 12.4i)T^{2} \) |
| 19 | \( 1 + (-1.61 - 2.79i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.32 + 2.76i)T + (15.6 - 16.8i)T^{2} \) |
| 29 | \( 1 + (-7.21 - 5.75i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-4.80 - 2.77i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.88 - 5.81i)T + (10.9 - 35.3i)T^{2} \) |
| 41 | \( 1 + (-0.287 + 0.0656i)T + (36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (1.83 - 1.15i)T + (18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-5.04 + 5.86i)T + (-7.00 - 46.4i)T^{2} \) |
| 53 | \( 1 + (5.86 + 4.32i)T + (15.6 + 50.6i)T^{2} \) |
| 59 | \( 1 + (-0.191 - 0.177i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-2.02 + 13.4i)T + (-58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (3.81 + 14.2i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.42 - 1.79i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (7.38 + 8.58i)T + (-10.8 + 72.1i)T^{2} \) |
| 79 | \( 1 + (0.349 - 0.201i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.729 - 2.08i)T + (-64.8 - 51.7i)T^{2} \) |
| 89 | \( 1 + (-10.0 - 6.82i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (0.336 + 0.336i)T + 97iT^{2} \) |
| 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
−12.44654898666288683042213798497, −11.74318001819131574354359139797, −10.64436043460220669083179492920, −10.19021200059813093779824859521, −8.067199223143602125582837612724, −7.02258067432298763862478014009, −6.46509077373288081609829328886, −5.04138953139676503352878641363, −4.72882096628659895411821159256, −3.34184275057942938585802943713,
1.06446453324471132460089064362, 2.65705043280469793530327619310, 4.58843989962820020979639717400, 5.23609611695457822198256324030, 5.90773815598649448538695736966, 7.51027271931853560194061927353, 8.843271156137765719179437172629, 10.30357003269412596799204486077, 11.41608011699467413421147496859, 11.72711299400878736932121952746
