| L(s) = 1 | + (0.170 − 0.0818i)2-s + (0.966 + 0.465i)3-s + (−4.96 + 6.22i)4-s + (−3.57 + 15.6i)5-s + 0.202·6-s + 29.7·7-s + (−0.670 + 2.93i)8-s + (−16.1 − 20.2i)9-s + (0.674 + 2.95i)10-s + (7.64 + 9.58i)11-s + (−7.70 + 3.70i)12-s + (−8.30 + 36.3i)13-s + (5.05 − 2.43i)14-s + (−10.7 + 13.4i)15-s + (−14.0 − 61.5i)16-s + (−7.89 − 34.6i)17-s + ⋯ |
| L(s) = 1 | + (0.0601 − 0.0289i)2-s + (0.186 + 0.0896i)3-s + (−0.620 + 0.778i)4-s + (−0.319 + 1.40i)5-s + 0.0137·6-s + 1.60·7-s + (−0.0296 + 0.129i)8-s + (−0.596 − 0.748i)9-s + (0.0213 + 0.0934i)10-s + (0.209 + 0.262i)11-s + (−0.185 + 0.0892i)12-s + (−0.177 + 0.775i)13-s + (0.0965 − 0.0464i)14-s + (−0.184 + 0.231i)15-s + (−0.219 − 0.961i)16-s + (−0.112 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.02286 + 0.788585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02286 + 0.788585i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (112. + 258. i)T \) |
| good | 2 | \( 1 + (-0.170 + 0.0818i)T + (4.98 - 6.25i)T^{2} \) |
| 3 | \( 1 + (-0.966 - 0.465i)T + (16.8 + 21.1i)T^{2} \) |
| 5 | \( 1 + (3.57 - 15.6i)T + (-112. - 54.2i)T^{2} \) |
| 7 | \( 1 - 29.7T + 343T^{2} \) |
| 11 | \( 1 + (-7.64 - 9.58i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (8.30 - 36.3i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (7.89 + 34.6i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 + (-59.7 + 74.9i)T + (-1.52e3 - 6.68e3i)T^{2} \) |
| 23 | \( 1 + (-76.9 - 96.4i)T + (-2.70e3 + 1.18e4i)T^{2} \) |
| 29 | \( 1 + (-175. + 84.4i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-57.7 + 27.7i)T + (1.85e4 - 2.32e4i)T^{2} \) |
| 37 | \( 1 - 286.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (398. - 191. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 47 | \( 1 + (289. - 362. i)T + (-2.31e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-31.6 - 138. i)T + (-1.34e5 + 6.45e4i)T^{2} \) |
| 59 | \( 1 + (110. + 483. i)T + (-1.85e5 + 8.91e4i)T^{2} \) |
| 61 | \( 1 + (130. + 62.8i)T + (1.41e5 + 1.77e5i)T^{2} \) |
| 67 | \( 1 + (-445. + 558. i)T + (-6.69e4 - 2.93e5i)T^{2} \) |
| 71 | \( 1 + (342. - 429. i)T + (-7.96e4 - 3.48e5i)T^{2} \) |
| 73 | \( 1 + (-114. + 501. i)T + (-3.50e5 - 1.68e5i)T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (113. + 54.8i)T + (3.56e5 + 4.47e5i)T^{2} \) |
| 89 | \( 1 + (69.6 + 33.5i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (125. + 156. i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
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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
−15.34858073798384204316599706449, −14.45540788696621497449132064527, −13.77585302360778637143019471804, −11.71511516720176864047557517353, −11.38876367921131736809943753201, −9.446978842113622178556392814715, −8.120583715289216827003618694516, −6.97392984352533779574000093776, −4.65146713772635234814234923091, −3.01991405472048188872665786232,
1.23226981068781841728907488093, 4.67240367722336961845485145805, 5.40460508543833329844237738160, 8.194729938968729713920731194186, 8.601210238642444287148149002627, 10.38180684822228796076027622617, 11.68463888082425666182392314208, 13.03964403290439303433125985618, 14.14583562005110581413691264188, 14.96356713036781761416502655690
