| L(s) = 1 | + (−1.15 − 0.821i)2-s + (−1.21 + 1.02i)3-s + (0.649 + 1.89i)4-s + (2.19 + 0.798i)5-s + (2.23 − 0.174i)6-s + (1.56 + 0.905i)7-s + (0.807 − 2.71i)8-s + (−0.0835 + 0.474i)9-s + (−1.86 − 2.72i)10-s + (1.01 − 0.586i)11-s + (−2.71 − 1.63i)12-s + (−2.13 + 2.54i)13-s + (−1.06 − 2.33i)14-s + (−3.48 + 1.26i)15-s + (−3.15 + 2.45i)16-s + (−1.18 − 6.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.813 − 0.581i)2-s + (−0.701 + 0.588i)3-s + (0.324 + 0.945i)4-s + (0.981 + 0.357i)5-s + (0.913 − 0.0713i)6-s + (0.593 + 0.342i)7-s + (0.285 − 0.958i)8-s + (−0.0278 + 0.158i)9-s + (−0.591 − 0.861i)10-s + (0.306 − 0.176i)11-s + (−0.784 − 0.472i)12-s + (−0.591 + 0.704i)13-s + (−0.283 − 0.623i)14-s + (−0.899 + 0.327i)15-s + (−0.789 + 0.613i)16-s + (−0.288 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.623487 + 0.130130i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.623487 + 0.130130i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.15 + 0.821i)T \) |
| 19 | \( 1 + (-2.39 - 3.64i)T \) |
| good | 3 | \( 1 + (1.21 - 1.02i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-2.19 - 0.798i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.56 - 0.905i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.01 + 0.586i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.13 - 2.54i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.18 + 6.74i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.21 + 3.32i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.974 + 0.171i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-5.07 + 8.78i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.26iT - 37T^{2} \) |
| 41 | \( 1 + (-6.74 - 8.03i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.85 - 7.83i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (6.49 + 1.14i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.859 + 2.36i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.0105 - 0.0596i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.43 + 0.887i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.731 - 4.14i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (14.6 + 5.32i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-5.56 + 4.66i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.92 + 2.45i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.29 - 4.20i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.45 - 5.31i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.53 + 1.68i)T + (91.1 - 33.1i)T^{2} \) |
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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
−14.53394373586473813678210581978, −13.45961141773634705269995387056, −11.80320599378446010801338245439, −11.31286261157677039301966047763, −10.01098105699882355485356005788, −9.446058277240838169781738966599, −7.84455159857082955926333331640, −6.25440287707868370662242360070, −4.68766360755894324982993430271, −2.34336253361731777710061078957,
1.48192582643336770647380694431, 5.20463074846649797638230486272, 6.20843884978937603614341392328, 7.35301481336082252296316987346, 8.718286060800591089623087025838, 9.909195783410548223087868960215, 10.94558594567170639607576723909, 12.20998077500035917273212440833, 13.44008577376314467332992879782, 14.56306962338961974083846787104
