| L(s) = 1 | + (0.0854 − 0.234i)2-s + (−2.26 + 0.399i)3-s + (1.48 + 1.24i)4-s + (−1.06 + 1.96i)5-s + (−0.0998 + 0.566i)6-s + (3.42 + 1.98i)7-s + (0.851 − 0.491i)8-s + (2.15 − 0.785i)9-s + (0.370 + 0.418i)10-s + (−1.56 − 2.71i)11-s + (−3.86 − 2.22i)12-s + (−2.61 − 0.461i)13-s + (0.757 − 0.635i)14-s + (1.63 − 4.88i)15-s + (0.630 + 3.57i)16-s + (−0.771 + 2.12i)17-s + ⋯ |
| L(s) = 1 | + (0.0604 − 0.165i)2-s + (−1.30 + 0.230i)3-s + (0.742 + 0.622i)4-s + (−0.476 + 0.879i)5-s + (−0.0407 + 0.231i)6-s + (1.29 + 0.748i)7-s + (0.301 − 0.173i)8-s + (0.719 − 0.261i)9-s + (0.117 + 0.132i)10-s + (−0.472 − 0.818i)11-s + (−1.11 − 0.643i)12-s + (−0.726 − 0.128i)13-s + (0.202 − 0.169i)14-s + (0.421 − 1.26i)15-s + (0.157 + 0.893i)16-s + (−0.187 + 0.514i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.709616 + 0.405320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.709616 + 0.405320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.06 - 1.96i)T \) |
| 19 | \( 1 + (-3.76 + 2.19i)T \) |
| good | 2 | \( 1 + (-0.0854 + 0.234i)T + (-1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (2.26 - 0.399i)T + (2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-3.42 - 1.98i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.56 + 2.71i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.61 + 0.461i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.771 - 2.12i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.87 + 5.81i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.28 + 0.466i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.447 - 0.774i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.62iT - 37T^{2} \) |
| 41 | \( 1 + (-1.08 - 6.14i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.13 + 1.35i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.0603 - 0.165i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (4.35 - 5.19i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (7.34 + 2.67i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.796 + 0.668i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.20 - 14.2i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-7.99 + 6.70i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (4.11 - 0.726i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.94 + 11.0i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (11.4 + 6.62i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.257 - 1.46i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.26 + 14.4i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−14.39096191209070814453460706625, −12.62366909779377511370147785488, −11.69697464193294823612235829141, −11.16376846269720710361366205498, −10.56599496896631172399687195623, −8.397073119865329930588636729556, −7.30241564668919568660436413722, −6.04271845284334353148862482145, −4.76327762653539722254209973791, −2.76211628820749971737688760440,
1.32065621918977940245200952704, 4.85756499236941150788311735603, 5.29244548082290166685267887270, 7.04743690611235267119561402316, 7.76403811709877009370842839158, 9.760297827070881598407993823150, 10.99254020612937720343214943235, 11.59100619664143016313156509497, 12.38973113170758259738560396183, 13.85111693471662815789645324110
