| L(s) = 1 | + (1.33 − 0.467i)2-s + (−0.558 − 4.96i)3-s + (1.56 − 1.24i)4-s + (−2.91 + 6.04i)5-s + (−3.06 − 6.36i)6-s + (4.81 − 6.03i)7-s + (1.50 − 2.39i)8-s + (−15.5 + 3.54i)9-s + (−1.06 + 9.43i)10-s + (5.19 + 8.26i)11-s + (−7.05 − 7.05i)12-s + (19.8 + 4.52i)13-s + (3.60 − 10.3i)14-s + (31.6 + 11.0i)15-s + (0.890 − 3.89i)16-s + (−13.5 + 13.5i)17-s + ⋯ |
| L(s) = 1 | + (0.667 − 0.233i)2-s + (−0.186 − 1.65i)3-s + (0.390 − 0.311i)4-s + (−0.582 + 1.20i)5-s + (−0.510 − 1.06i)6-s + (0.687 − 0.862i)7-s + (0.188 − 0.299i)8-s + (−1.72 + 0.393i)9-s + (−0.106 + 0.943i)10-s + (0.472 + 0.751i)11-s + (−0.588 − 0.588i)12-s + (1.52 + 0.347i)13-s + (0.257 − 0.735i)14-s + (2.10 + 0.737i)15-s + (0.0556 − 0.243i)16-s + (−0.797 + 0.797i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.19153 - 0.955921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19153 - 0.955921i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 + 0.467i)T \) |
| 29 | \( 1 + (-8.87 + 27.6i)T \) |
| good | 3 | \( 1 + (0.558 + 4.96i)T + (-8.77 + 2.00i)T^{2} \) |
| 5 | \( 1 + (2.91 - 6.04i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 + (-4.81 + 6.03i)T + (-10.9 - 47.7i)T^{2} \) |
| 11 | \( 1 + (-5.19 - 8.26i)T + (-52.4 + 109. i)T^{2} \) |
| 13 | \( 1 + (-19.8 - 4.52i)T + (152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (13.5 - 13.5i)T - 289iT^{2} \) |
| 19 | \( 1 + (16.3 + 1.84i)T + (351. + 80.3i)T^{2} \) |
| 23 | \( 1 + (2.47 - 1.19i)T + (329. - 413. i)T^{2} \) |
| 31 | \( 1 + (39.3 - 13.7i)T + (751. - 599. i)T^{2} \) |
| 37 | \( 1 + (17.4 - 27.7i)T + (-593. - 1.23e3i)T^{2} \) |
| 41 | \( 1 + (-5.70 - 5.70i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-18.8 + 53.9i)T + (-1.44e3 - 1.15e3i)T^{2} \) |
| 47 | \( 1 + (37.3 - 23.4i)T + (958. - 1.99e3i)T^{2} \) |
| 53 | \( 1 + (8.97 + 4.32i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 - 21.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + (9.69 + 86.0i)T + (-3.62e3 + 828. i)T^{2} \) |
| 67 | \( 1 + (-66.2 + 15.1i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-26.7 - 6.09i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (23.2 + 8.13i)T + (4.16e3 + 3.32e3i)T^{2} \) |
| 79 | \( 1 + (-44.4 - 27.9i)T + (2.70e3 + 5.62e3i)T^{2} \) |
| 83 | \( 1 + (-33.1 - 41.5i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (127. - 44.5i)T + (6.19e3 - 4.93e3i)T^{2} \) |
| 97 | \( 1 + (19.4 - 172. i)T + (-9.17e3 - 2.09e3i)T^{2} \) |
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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
−14.32726251751923327846175614184, −13.58030952530955789804366855565, −12.53538089207946015755867535358, −11.30650862871011182706674619520, −10.83622010327270371317277447294, −8.180600966214167945575756939091, −6.99529360908554076886630524601, −6.39331270611461636047789913195, −3.93549581021981710437633976504, −1.80765256299452518221139545817,
3.75338256048516801255937575543, 4.80788079235937435581206078429, 5.80967559150555820972094689401, 8.536313792516358365300646386945, 8.941352615524986889283087169873, 10.95075285897414143163323344743, 11.56666304562266115837808621879, 12.91786845752363175006279488626, 14.38961343368066209026714258351, 15.40312121164099131675728171690
