| L(s) = 1 | + (−3.61 − 0.316i)2-s + (−3.31 − 4.00i)3-s + (5.08 + 0.896i)4-s + (−10.5 + 3.67i)5-s + (10.7 + 15.5i)6-s + (18.3 + 12.8i)7-s + (9.94 + 2.66i)8-s + (−5.03 + 26.5i)9-s + (39.3 − 9.92i)10-s + (19.2 − 52.8i)11-s + (−13.2 − 23.3i)12-s + (4.33 + 49.5i)13-s + (−62.1 − 52.1i)14-s + (49.6 + 30.0i)15-s + (−73.9 − 26.8i)16-s + (−88.4 + 23.6i)17-s + ⋯ |
| L(s) = 1 | + (−1.27 − 0.111i)2-s + (−0.637 − 0.770i)3-s + (0.635 + 0.112i)4-s + (−0.944 + 0.328i)5-s + (0.728 + 1.05i)6-s + (0.989 + 0.692i)7-s + (0.439 + 0.117i)8-s + (−0.186 + 0.982i)9-s + (1.24 − 0.313i)10-s + (0.526 − 1.44i)11-s + (−0.319 − 0.561i)12-s + (0.0925 + 1.05i)13-s + (−1.18 − 0.995i)14-s + (0.855 + 0.518i)15-s + (−1.15 − 0.420i)16-s + (−1.26 + 0.338i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.276805 - 0.331865i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.276805 - 0.331865i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.31 + 4.00i)T \) |
| 5 | \( 1 + (10.5 - 3.67i)T \) |
| good | 2 | \( 1 + (3.61 + 0.316i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-18.3 - 12.8i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (-19.2 + 52.8i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-4.33 - 49.5i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (88.4 - 23.6i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (-36.7 + 21.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (45.8 + 65.4i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (38.2 - 32.1i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-48.3 + 274. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-7.51 - 28.0i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-100. + 119. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-23.9 - 51.2i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (-299. + 427. i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (250. - 250. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-140. + 51.2i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (26.9 + 153. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-385. + 33.7i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (392. + 226. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-51.4 + 191. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (826. + 984. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-102. + 1.16e3i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-319. - 552. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.00e3 + 470. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−11.79892455920113528880306337241, −11.41370339540094337523467746016, −10.76068710968145494145183282847, −8.911644730068909695213100352173, −8.365464649358450724024728451872, −7.35943592090710097624572899271, −6.18776735117594863635619341446, −4.48490146963582777602466848431, −2.04767464064051601307879537808, −0.45951972693137869376637698728,
1.09517378939291522744523369182, 4.12436560441128926118295647384, 4.89524941322796280157517742996, 7.00446139516630037381893333968, 7.84021786987959351015682479686, 8.924255917315417131892945528908, 9.934503272576204309174043505601, 10.83238923911412219783828273887, 11.56203849425319159550522380147, 12.70507258573808908022684412187
