L(s) = 1 | + (0.882 + 3.86i)2-s + (4.09 + 3.79i)3-s + (−6.96 + 3.35i)4-s + (7.73 − 1.16i)5-s + (−11.0 + 19.1i)6-s + (−13.6 − 23.6i)7-s + (0.667 + 0.836i)8-s + (0.313 + 4.17i)9-s + (11.3 + 28.8i)10-s + (−32.0 − 15.4i)11-s + (−41.2 − 12.7i)12-s + (−4.10 + 10.4i)13-s + (79.3 − 73.5i)14-s + (36.0 + 24.5i)15-s + (−41.2 + 51.6i)16-s + (97.9 + 14.7i)17-s + ⋯ |
L(s) = 1 | + (0.312 + 1.36i)2-s + (0.787 + 0.731i)3-s + (−0.870 + 0.419i)4-s + (0.691 − 0.104i)5-s + (−0.753 + 1.30i)6-s + (−0.736 − 1.27i)7-s + (0.0294 + 0.0369i)8-s + (0.0116 + 0.154i)9-s + (0.358 + 0.912i)10-s + (−0.878 − 0.423i)11-s + (−0.992 − 0.306i)12-s + (−0.0876 + 0.223i)13-s + (1.51 − 1.40i)14-s + (0.620 + 0.423i)15-s + (−0.643 + 0.807i)16-s + (1.39 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.08018 + 1.57056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08018 + 1.57056i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-249. + 131. i)T \) |
good | 2 | \( 1 + (-0.882 - 3.86i)T + (-7.20 + 3.47i)T^{2} \) |
| 3 | \( 1 + (-4.09 - 3.79i)T + (2.01 + 26.9i)T^{2} \) |
| 5 | \( 1 + (-7.73 + 1.16i)T + (119. - 36.8i)T^{2} \) |
| 7 | \( 1 + (13.6 + 23.6i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (32.0 + 15.4i)T + (829. + 1.04e3i)T^{2} \) |
| 13 | \( 1 + (4.10 - 10.4i)T + (-1.61e3 - 1.49e3i)T^{2} \) |
| 17 | \( 1 + (-97.9 - 14.7i)T + (4.69e3 + 1.44e3i)T^{2} \) |
| 19 | \( 1 + (-0.481 + 6.42i)T + (-6.78e3 - 1.02e3i)T^{2} \) |
| 23 | \( 1 + (-51.0 + 34.8i)T + (4.44e3 - 1.13e4i)T^{2} \) |
| 29 | \( 1 + (186. - 173. i)T + (1.82e3 - 2.43e4i)T^{2} \) |
| 31 | \( 1 + (-38.7 - 11.9i)T + (2.46e4 + 1.67e4i)T^{2} \) |
| 37 | \( 1 + (147. - 255. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (47.9 + 209. i)T + (-6.20e4 + 2.99e4i)T^{2} \) |
| 47 | \( 1 + (308. - 148. i)T + (6.47e4 - 8.11e4i)T^{2} \) |
| 53 | \( 1 + (132. + 337. i)T + (-1.09e5 + 1.01e5i)T^{2} \) |
| 59 | \( 1 + (-229. + 287. i)T + (-4.57e4 - 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-750. + 231. i)T + (1.87e5 - 1.27e5i)T^{2} \) |
| 67 | \( 1 + (-27.9 + 372. i)T + (-2.97e5 - 4.48e4i)T^{2} \) |
| 71 | \( 1 + (537. + 366. i)T + (1.30e5 + 3.33e5i)T^{2} \) |
| 73 | \( 1 + (385. - 981. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-431. - 747. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-389. - 361. i)T + (4.27e4 + 5.70e5i)T^{2} \) |
| 89 | \( 1 + (-793. - 736. i)T + (5.26e4 + 7.02e5i)T^{2} \) |
| 97 | \( 1 + (487. + 234. i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−15.85717577518342764429598530265, −14.65741581684750264651185885534, −13.92840103352446207943899911032, −13.04698571900475060324304942740, −10.54445279159540700965107773073, −9.547399669692846802397564283511, −8.112529918162159072504284717557, −6.82374550803035430124345744552, −5.33324695774196559329775381531, −3.59916009190537388528736855325,
2.07589585139863515901110272956, 3.00473020310611555921518358548, 5.58791474746059120878393231371, 7.61689940023678363593611009506, 9.299616555310810439497545054281, 10.21183756024858223170720597121, 11.86108970363494943664435849872, 12.85615883607116161936258788754, 13.38313893367819279155763470663, 14.68783524657529618336258026073