| L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)6-s + (1.43 − 2.49i)7-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.939 − 0.342i)10-s + (2.26 + 3.92i)11-s + (0.5 − 0.866i)12-s + (0.368 − 2.08i)13-s + (2.20 − 1.85i)14-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (−4.43 − 1.61i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.100 − 0.568i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (0.0708 − 0.402i)6-s + (0.544 − 0.942i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (0.297 − 0.108i)10-s + (0.683 + 1.18i)11-s + (0.144 − 0.249i)12-s + (0.102 − 0.579i)13-s + (0.589 − 0.494i)14-s + (−0.197 − 0.165i)15-s + (0.0434 + 0.246i)16-s + (−1.07 − 0.391i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.23232 - 0.662595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23232 - 0.662595i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-3.37 + 2.75i)T \) |
| good | 7 | \( 1 + (-1.43 + 2.49i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.26 - 3.92i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.368 + 2.08i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (4.43 + 1.61i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (2.20 + 1.85i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-9.33 + 3.39i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.0714 - 0.123i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 + (-1.50 - 8.52i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.44 - 6.24i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.35 + 1.58i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-10.5 - 8.83i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (13.1 + 4.77i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.42 - 2.87i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (7.23 - 2.63i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (10.8 - 9.11i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.28 + 12.9i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.481 + 2.73i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.488 - 0.846i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.45 - 13.9i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.634 - 0.230i)T + (74.3 + 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77858162764993624079528120325, −9.888830065060198279914394146124, −8.735327994065900201448611725132, −7.70610582752992023661965557423, −6.98400499419173332595330957444, −6.21983906298267356357455455449, −4.86522422544245164776163218727, −4.33703696528672797625567549490, −2.69274044259346828405581098650, −1.30008748831304633294229346016,
1.80810692912743764914209796684, 3.12285594160893025226368868776, 4.13750692796181119544317423207, 5.31048787589344637787960122200, 5.97706200395853854751860786093, 6.90165593841709634934953953555, 8.530249643727824181612039862529, 8.964577273403923112040998670125, 10.17785280795969232179926705670, 10.90920162624625751206232125355
