| L(s) = 1 | + (0.791 − 0.288i)2-s + (0.173 − 0.984i)3-s + (−0.988 + 0.829i)4-s + (1.30 + 1.09i)5-s + (−0.146 − 0.829i)6-s + (−1.96 − 3.40i)7-s + (−1.38 + 2.40i)8-s + (−0.939 − 0.342i)9-s + (1.35 + 0.491i)10-s + (−2.21 + 3.83i)11-s + (0.645 + 1.11i)12-s + (0.316 + 1.79i)13-s + (−2.53 − 2.12i)14-s + (1.30 − 1.09i)15-s + (0.0427 − 0.242i)16-s + (4.72 − 1.71i)17-s + ⋯ |
| L(s) = 1 | + (0.559 − 0.203i)2-s + (0.100 − 0.568i)3-s + (−0.494 + 0.414i)4-s + (0.584 + 0.490i)5-s + (−0.0597 − 0.338i)6-s + (−0.742 − 1.28i)7-s + (−0.489 + 0.848i)8-s + (−0.313 − 0.114i)9-s + (0.427 + 0.155i)10-s + (−0.667 + 1.15i)11-s + (0.186 + 0.322i)12-s + (0.0878 + 0.498i)13-s + (−0.677 − 0.568i)14-s + (0.337 − 0.283i)15-s + (0.0106 − 0.0606i)16-s + (1.14 − 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00248 - 0.176025i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00248 - 0.176025i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (1.33 + 4.14i)T \) |
| good | 2 | \( 1 + (-0.791 + 0.288i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-1.30 - 1.09i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.96 + 3.40i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.21 - 3.83i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.316 - 1.79i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.72 + 1.71i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.64 + 2.22i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.36 - 0.496i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.43 - 2.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.83T + 37T^{2} \) |
| 41 | \( 1 + (1.52 - 8.67i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.73 - 6.49i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (3.22 + 1.17i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.61 - 1.35i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (6.01 - 2.19i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.0587 + 0.0492i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.984 - 0.358i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (10.3 + 8.70i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.696 + 3.95i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.94 + 16.7i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.51 - 11.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.14 - 12.1i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (0.164 - 0.0599i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.69141840325497457311638849211, −13.79498037781045261033483126832, −13.12527547061852588617173669464, −12.18935030192250355234539710449, −10.56523865867394009946248785734, −9.449957502680771828418908166708, −7.68785450022252566950695810763, −6.60455271361392132659682100790, −4.67684905685693633391198361335, −2.93907399245567344955968053846,
3.37691583669626984376694854077, 5.52794438712142401363546741461, 5.76475615525760624793541246028, 8.491065190943209323470972783342, 9.426464813992807188446453707241, 10.43372321821675053654262988367, 12.30076375218959713733751661292, 13.16586541627607638783511463991, 14.19534836222936866250100612311, 15.33467597251693912411912511809
