| L(s) = 1 | + (−0.819 + 0.573i)2-s + (−0.0360 − 1.73i)3-s + (0.342 − 0.939i)4-s + (0.368 + 2.20i)5-s + (1.02 + 1.39i)6-s + (−0.623 − 0.166i)7-s + (0.258 + 0.965i)8-s + (−2.99 + 0.125i)9-s + (−1.56 − 1.59i)10-s + (−1.95 − 1.12i)11-s + (−1.63 − 0.558i)12-s + (−3.88 − 0.340i)13-s + (0.606 − 0.220i)14-s + (3.80 − 0.717i)15-s + (−0.766 − 0.642i)16-s + (−4.32 + 3.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 + 0.405i)2-s + (−0.0208 − 0.999i)3-s + (0.171 − 0.469i)4-s + (0.164 + 0.986i)5-s + (0.417 + 0.570i)6-s + (−0.235 − 0.0630i)7-s + (0.0915 + 0.341i)8-s + (−0.999 + 0.0416i)9-s + (−0.495 − 0.504i)10-s + (−0.588 − 0.339i)11-s + (−0.473 − 0.161i)12-s + (−1.07 − 0.0943i)13-s + (0.161 − 0.0589i)14-s + (0.982 − 0.185i)15-s + (−0.191 − 0.160i)16-s + (−1.04 + 0.735i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00144490 - 0.0612468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00144490 - 0.0612468i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 - 0.573i)T \) |
| 3 | \( 1 + (0.0360 + 1.73i)T \) |
| 5 | \( 1 + (-0.368 - 2.20i)T \) |
| 19 | \( 1 + (-0.0260 + 4.35i)T \) |
| good | 7 | \( 1 + (0.623 + 0.166i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.95 + 1.12i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.88 + 0.340i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (4.32 - 3.03i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (-1.63 + 0.760i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (1.38 + 7.84i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.69 - 2.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.860 + 0.860i)T - 37iT^{2} \) |
| 41 | \( 1 + (3.79 - 4.52i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (8.64 + 4.02i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (6.88 - 9.82i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (6.10 - 2.84i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (0.503 - 2.85i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-9.92 - 3.61i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.903 + 0.632i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-0.913 - 2.51i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (0.670 + 7.65i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (-6.38 + 7.60i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.78 - 0.746i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-8.24 + 6.92i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.15 - 10.2i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−10.31497887448004469139554502226, −9.377280552808140185372641528723, −8.294273962890328365971599190546, −7.57050288469515684453628376199, −6.70249764306347439728531965279, −6.22054015649811320238334188513, −4.94831624648352777927575238243, −3.00318801748458050044804215915, −2.09730983242718478436404747433, −0.03815841844299119662337670408,
2.10673072168970170237728063365, 3.42438627486341817827923550605, 4.72209216112666115654721494242, 5.26036066413126972934322332015, 6.74729375497551975852578093495, 8.008275747518589920798988043615, 8.745241171300655017033694082840, 9.678306262813770178143454796369, 9.928405745377217840652247294981, 11.05032978028888446790166079140
