| 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
−13.85111693471662815789645324110, −12.38973113170758259738560396183, −11.59100619664143016313156509497, −10.99254020612937720343214943235, −9.760297827070881598407993823150, −7.76403811709877009370842839158, −7.04743690611235267119561402316, −5.29244548082290166685267887270, −4.85756499236941150788311735603, −1.32065621918977940245200952704,
2.76211628820749971737688760440, 4.76327762653539722254209973791, 6.04271845284334353148862482145, 7.30241564668919568660436413722, 8.397073119865329930588636729556, 10.56599496896631172399687195623, 11.16376846269720710361366205498, 11.69697464193294823612235829141, 12.62366909779377511370147785488, 14.39096191209070814453460706625
