| L(s) = 1 | + (0.0268 − 0.0738i)2-s + (1.70 − 0.300i)3-s + (3.05 + 2.56i)4-s + (−2.91 + 2.44i)5-s + (0.0236 − 0.134i)6-s + (2.84 − 4.92i)7-s + (0.544 − 0.314i)8-s + (2.81 − 1.02i)9-s + (0.102 + 0.280i)10-s + (−1.02 − 1.78i)11-s + (5.99 + 3.45i)12-s + (−4.27 − 0.754i)13-s + (−0.287 − 0.342i)14-s + (−4.23 + 5.04i)15-s + (2.76 + 15.6i)16-s + (−28.2 − 10.2i)17-s + ⋯ |
| L(s) = 1 | + (0.0134 − 0.0369i)2-s + (0.568 − 0.100i)3-s + (0.764 + 0.641i)4-s + (−0.582 + 0.488i)5-s + (0.00394 − 0.0223i)6-s + (0.406 − 0.703i)7-s + (0.0680 − 0.0392i)8-s + (0.313 − 0.114i)9-s + (0.0102 + 0.0280i)10-s + (−0.0935 − 0.161i)11-s + (0.499 + 0.288i)12-s + (−0.329 − 0.0580i)13-s + (−0.0205 − 0.0244i)14-s + (−0.282 + 0.336i)15-s + (0.172 + 0.980i)16-s + (−1.66 − 0.604i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44032 + 0.164610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44032 + 0.164610i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 + 0.300i)T \) |
| 19 | \( 1 + (6.47 + 17.8i)T \) |
| good | 2 | \( 1 + (-0.0268 + 0.0738i)T + (-3.06 - 2.57i)T^{2} \) |
| 5 | \( 1 + (2.91 - 2.44i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-2.84 + 4.92i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (1.02 + 1.78i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.27 + 0.754i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (28.2 + 10.2i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (1.23 + 1.03i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-5.64 - 15.5i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-13.3 - 7.68i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 25.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-66.2 + 11.6i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (30.1 - 25.2i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-48.3 + 17.6i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-33.1 + 39.4i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (36.3 - 99.9i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (81.8 + 68.6i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-12.3 - 33.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-62.5 - 74.5i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (0.571 + 3.23i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (60.6 - 10.6i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (30.0 - 51.9i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (33.0 + 5.83i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (40.0 - 109. i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−15.17398637619904936267868304254, −13.89583947145818138336346330004, −12.81304841116464648904709351910, −11.42884887665675744293534051154, −10.74380042676046404546223945711, −8.865306652808231105532244921361, −7.56073069786773962693004114123, −6.85721761344985547861586691938, −4.22355376656891807458083512382, −2.65158685292804053544618195082,
2.20054564544889967155652952791, 4.50967778815408424315305728414, 6.19456528486971014153657729734, 7.79446406858798752572567342307, 8.905477882905784186529153263065, 10.32644462082831233387577237562, 11.54895003011216027569576388774, 12.55271245418013508286843580337, 14.06097062190735182221171360107, 15.25474792186279197543594779743
