| L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.434 − 1.19i)3-s + (0.766 − 0.642i)4-s + (−0.640 + 2.14i)5-s + 1.27i·6-s + (−2.39 + 0.422i)7-s + (−0.500 + 0.866i)8-s + (1.06 + 0.890i)9-s + (−0.131 − 2.23i)10-s + (−0.539 + 0.934i)11-s + (−0.434 − 1.19i)12-s + (−4.50 + 3.77i)13-s + (2.10 − 1.21i)14-s + (2.28 + 1.69i)15-s + (0.173 − 0.984i)16-s + (−3.03 − 2.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.250 − 0.689i)3-s + (0.383 − 0.321i)4-s + (−0.286 + 0.958i)5-s + 0.518i·6-s + (−0.905 + 0.159i)7-s + (−0.176 + 0.306i)8-s + (0.353 + 0.296i)9-s + (−0.0414 − 0.705i)10-s + (−0.162 + 0.281i)11-s + (−0.125 − 0.344i)12-s + (−1.24 + 1.04i)13-s + (0.563 − 0.325i)14-s + (0.588 + 0.437i)15-s + (0.0434 − 0.246i)16-s + (−0.734 − 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.361923 + 0.519874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.361923 + 0.519874i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.640 - 2.14i)T \) |
| 37 | \( 1 + (5.80 - 1.82i)T \) |
| good | 3 | \( 1 + (-0.434 + 1.19i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (2.39 - 0.422i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.539 - 0.934i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.50 - 3.77i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.03 + 2.54i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (0.238 - 0.656i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-4.52 - 7.83i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.50 - 2.02i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.83iT - 31T^{2} \) |
| 41 | \( 1 + (4.58 - 3.85i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 - 2.24T + 43T^{2} \) |
| 47 | \( 1 + (-6.62 + 3.82i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.71 - 1.36i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (8.83 + 1.55i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.74 + 9.22i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (4.03 - 0.710i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-10.0 - 3.67i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 0.0962iT - 73T^{2} \) |
| 79 | \( 1 + (0.293 - 0.0516i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-9.17 + 10.9i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-12.0 - 2.12i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.60 - 4.50i)T + (-48.5 + 84.0i)T^{2} \) |
| 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
−11.68171723707720241146495411471, −10.59008468563570960673923224078, −9.774108184079166286101324847975, −8.983740652547977711968640050119, −7.60849819628201071351873925672, −7.06205759338278183549555891998, −6.53344199487270709442090487272, −4.88020497466383837254453574084, −3.11828333166615720253267372466, −2.01375947822578231962660925202,
0.49237031310594058734424306409, 2.71247065617395787470411088310, 3.95390512714756316756667632697, 4.98596923425306747131273998722, 6.46139675950426014871470528728, 7.57034681199208581807014502937, 8.674157280257454191782290114043, 9.232327166272742055721717557900, 10.18727391172882033836113320647, 10.68305007300790554636631308408
