| L(s) = 1 | + (−2.39 − 0.869i)2-s + (0.413 + 1.68i)3-s + (3.42 + 2.87i)4-s + (−1.33 + 1.79i)5-s + (0.473 − 4.37i)6-s + (−1.18 − 2.36i)7-s + (−3.13 − 5.43i)8-s + (−2.65 + 1.39i)9-s + (4.75 − 3.11i)10-s + (0.730 − 0.128i)11-s + (−3.41 + 6.94i)12-s + (4.21 − 1.53i)13-s + (0.760 + 6.68i)14-s + (−3.56 − 1.50i)15-s + (1.22 + 6.92i)16-s + (−1.33 − 0.770i)17-s + ⋯ |
| L(s) = 1 | + (−1.69 − 0.615i)2-s + (0.238 + 0.971i)3-s + (1.71 + 1.43i)4-s + (−0.598 + 0.801i)5-s + (0.193 − 1.78i)6-s + (−0.446 − 0.894i)7-s + (−1.11 − 1.92i)8-s + (−0.885 + 0.464i)9-s + (1.50 − 0.986i)10-s + (0.220 − 0.0388i)11-s + (−0.985 + 2.00i)12-s + (1.16 − 0.425i)13-s + (0.203 + 1.78i)14-s + (−0.921 − 0.389i)15-s + (0.305 + 1.73i)16-s + (−0.323 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.621564 + 0.0793300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.621564 + 0.0793300i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.413 - 1.68i)T \) |
| 5 | \( 1 + (1.33 - 1.79i)T \) |
| 7 | \( 1 + (1.18 + 2.36i)T \) |
| good | 2 | \( 1 + (2.39 + 0.869i)T + (1.53 + 1.28i)T^{2} \) |
| 11 | \( 1 + (-0.730 + 0.128i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.21 + 1.53i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.33 + 0.770i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.23 + 1.28i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.69 - 3.10i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.66 + 4.57i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.24 + 6.25i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-4.43 - 2.55i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.73 + 2.81i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (9.27 - 1.63i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.190 - 0.226i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 + (1.47 - 8.34i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.03 - 2.43i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.52 - 12.4i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (0.774 + 0.447i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.05 - 12.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-14.0 - 5.10i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.88 + 7.92i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (3.71 + 6.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.473 + 2.68i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.928324501976031844182044981032, −9.569847710408763384045526059466, −8.503928614950164661219117550305, −7.908546283449747704743497562302, −7.08329084045106267035239515313, −6.12345623447772660705137634119, −4.27231696384961438550676066027, −3.39418268785457297656329677978, −2.68667172790183898256358039128, −0.77411276917568679612883641249,
0.830485510739678289562929437375, 1.82286537708668005408558346925, 3.27640351911523784480045428791, 5.15272070740143432247795111262, 6.38562811086236561101246593573, 6.62145175781911057093194545115, 7.88174802313504658380381381059, 8.310775515479100354170464171766, 9.056783111052123857351863079427, 9.363126043902386272795578802677
