| L(s) = 1 | + (0.529 + 1.62i)2-s + (3.53 − 0.751i)3-s + (4.09 − 2.97i)4-s + (−3.32 + 5.75i)5-s + (3.09 + 5.36i)6-s + (−2.35 − 1.04i)7-s + (18.1 + 13.1i)8-s + (−12.7 + 5.66i)9-s + (−11.1 − 2.36i)10-s + (−1.78 − 16.9i)11-s + (12.2 − 13.6i)12-s + (−53.2 − 59.0i)13-s + (0.460 − 4.38i)14-s + (−7.42 + 22.8i)15-s + (0.684 − 2.10i)16-s + (−1.58 + 15.0i)17-s + ⋯ |
| L(s) = 1 | + (0.187 + 0.575i)2-s + (0.680 − 0.144i)3-s + (0.512 − 0.372i)4-s + (−0.297 + 0.514i)5-s + (0.210 + 0.364i)6-s + (−0.126 − 0.0565i)7-s + (0.800 + 0.581i)8-s + (−0.471 + 0.209i)9-s + (−0.352 − 0.0748i)10-s + (−0.0488 − 0.464i)11-s + (0.294 − 0.327i)12-s + (−1.13 − 1.26i)13-s + (0.00879 − 0.0836i)14-s + (−0.127 + 0.393i)15-s + (0.0106 − 0.0328i)16-s + (−0.0226 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.56717 + 0.409958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56717 + 0.409958i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-150. + 84.1i)T \) |
| good | 2 | \( 1 + (-0.529 - 1.62i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (-3.53 + 0.751i)T + (24.6 - 10.9i)T^{2} \) |
| 5 | \( 1 + (3.32 - 5.75i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (2.35 + 1.04i)T + (229. + 254. i)T^{2} \) |
| 11 | \( 1 + (1.78 + 16.9i)T + (-1.30e3 + 276. i)T^{2} \) |
| 13 | \( 1 + (53.2 + 59.0i)T + (-229. + 2.18e3i)T^{2} \) |
| 17 | \( 1 + (1.58 - 15.0i)T + (-4.80e3 - 1.02e3i)T^{2} \) |
| 19 | \( 1 + (13.4 - 14.9i)T + (-716. - 6.82e3i)T^{2} \) |
| 23 | \( 1 + (-26.6 - 19.3i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-32.2 - 99.3i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 37 | \( 1 + (58.3 + 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-164. - 35.0i)T + (6.29e4 + 2.80e4i)T^{2} \) |
| 43 | \( 1 + (149. - 166. i)T + (-8.31e3 - 7.90e4i)T^{2} \) |
| 47 | \( 1 + (62.7 - 193. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-614. + 273. i)T + (9.96e4 - 1.10e5i)T^{2} \) |
| 59 | \( 1 + (-413. + 87.8i)T + (1.87e5 - 8.35e4i)T^{2} \) |
| 61 | \( 1 - 671.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (328. - 568. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (694. - 309. i)T + (2.39e5 - 2.65e5i)T^{2} \) |
| 73 | \( 1 + (55.1 + 524. i)T + (-3.80e5 + 8.08e4i)T^{2} \) |
| 79 | \( 1 + (-41.4 + 394. i)T + (-4.82e5 - 1.02e5i)T^{2} \) |
| 83 | \( 1 + (1.32e3 + 281. i)T + (5.22e5 + 2.32e5i)T^{2} \) |
| 89 | \( 1 + (197. - 143. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (616. - 447. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−16.29599582059180594256897512944, −14.98892871381719810439011836530, −14.55052031681946268857015839897, −13.17080429488470878035397044953, −11.40396450481856313220519429415, −10.20259109044421167448916347233, −8.237599756094703591996203044016, −7.15670213088204093306993343443, −5.49854787617024702715441159540, −2.83066827285017180180191303412,
2.53882580742502552871981997648, 4.35464063025315605443188786407, 6.99158554408101280767871961051, 8.518040255363325638509473349231, 9.893838610786182870904259900790, 11.63169561821987569269858645476, 12.37724611093494553292636164319, 13.81274829945370089680985638321, 15.06964614550458563809215439797, 16.30902555741157731841040305581
