| L(s) = 1 | + (−1.43 + 0.253i)2-s + (1.11 + 1.32i)3-s + (−1.76 + 0.641i)4-s + (3.63 + 1.32i)5-s + (−1.93 − 1.62i)6-s + (−6.51 + 11.2i)7-s + (7.41 − 4.28i)8-s + (−0.520 + 2.95i)9-s + (−5.55 − 0.980i)10-s + (4.25 + 7.37i)11-s + (−2.81 − 1.62i)12-s + (2.30 − 2.74i)13-s + (6.49 − 17.8i)14-s + (2.29 + 6.30i)15-s + (−3.81 + 3.20i)16-s + (−4.38 − 24.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.717 + 0.126i)2-s + (0.371 + 0.442i)3-s + (−0.440 + 0.160i)4-s + (0.727 + 0.264i)5-s + (−0.322 − 0.270i)6-s + (−0.931 + 1.61i)7-s + (0.927 − 0.535i)8-s + (−0.0578 + 0.328i)9-s + (−0.555 − 0.0980i)10-s + (0.387 + 0.670i)11-s + (−0.234 − 0.135i)12-s + (0.177 − 0.211i)13-s + (0.464 − 1.27i)14-s + (0.152 + 0.420i)15-s + (−0.238 + 0.200i)16-s + (−0.258 − 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0226 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0226 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.597640 + 0.584271i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.597640 + 0.584271i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.11 - 1.32i)T \) |
| 19 | \( 1 + (-16.8 + 8.73i)T \) |
| good | 2 | \( 1 + (1.43 - 0.253i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-3.63 - 1.32i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (6.51 - 11.2i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.25 - 7.37i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-2.30 + 2.74i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (4.38 + 24.8i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (-6.42 + 2.33i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-43.3 - 7.63i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (0.623 + 0.359i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 61.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-3.33 - 3.97i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-23.2 - 8.47i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-4.46 + 25.3i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (24.1 + 66.2i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (0.148 - 0.0261i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-44.1 + 16.0i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-38.6 - 6.81i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-7.56 + 20.7i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (71.8 - 60.3i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-5.69 - 6.79i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (39.5 - 68.5i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-79.0 + 94.1i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (89.9 - 15.8i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−15.52934250807129995629224706569, −14.13549944188462222713797019584, −13.13121994378861731124488845133, −11.86247658822809949792012403443, −9.900335972293718720038570976878, −9.494143243969968837098448286390, −8.520663193029994174707041908135, −6.74656483066337174446269065117, −5.06495340780798690276987649546, −2.82214800883434535919442191387,
1.13120674910371827100137747031, 3.91296556487876235318552757161, 6.11921234875003240456807038114, 7.58569703059080090773624284760, 8.914661850170732232774918600268, 9.898346917121866066392413761985, 10.79151073579717300677952698664, 12.80208399816825568127732788489, 13.70171531220593266918204912047, 14.16550123089302797651288324180
