| L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−1.16 − 1.29i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.913 − 0.406i)10-s + (−4.27 + 0.909i)11-s + (−0.104 − 0.994i)12-s + (0.507 − 4.82i)13-s + (1.70 + 0.362i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−5.11 − 1.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.527 − 0.234i)3-s + (0.154 − 0.475i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.440 − 0.489i)7-s + (0.109 + 0.336i)8-s + (0.223 − 0.247i)9-s + (−0.288 − 0.128i)10-s + (−1.29 + 0.274i)11-s + (−0.0301 − 0.287i)12-s + (0.140 − 1.33i)13-s + (0.455 + 0.0967i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (−1.24 − 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.419691 - 0.577829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.419691 - 0.577829i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (4.96 + 2.52i)T \) |
| good | 7 | \( 1 + (1.16 + 1.29i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (4.27 - 0.909i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.507 + 4.82i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (5.11 + 1.08i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.132 + 1.26i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.898 + 2.76i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.88 + 1.37i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.26 + 3.91i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.29 + 1.02i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.805 - 7.66i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-5.28 - 3.84i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.00 + 4.45i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (11.8 - 5.29i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 5.21T + 61T^{2} \) |
| 67 | \( 1 + (5.26 + 9.12i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.42 + 1.58i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-1.54 + 0.327i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-1.48 - 0.314i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-8.73 - 3.89i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-0.697 + 2.14i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.20 + 16.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.767928658036384477324257407218, −8.951061593256094507269490420368, −7.970240575410501903425202268660, −7.47270006626973262318933221061, −6.58196469565083645520734316517, −5.69037025364922529197332709217, −4.53909287449211687262503361047, −3.09445709411464224768095226949, −2.23789801943229518223080977394, −0.35033497160388401345084697559,
1.81060648204519895512584246683, 2.69860869677908666676660045434, 3.87255925463926585358634375290, 4.92680463850808032734202905096, 6.07537688679645806147090979678, 7.10787660514741461163549080853, 8.080816983921197714683126546825, 8.955093394622256939681072180660, 9.219094758507777629912891896888, 10.28309132848284640090020363381
