| L(s) = 1 | + (0.116 − 1.40i)2-s + (1.65 − 2.36i)3-s + (−1.97 − 0.328i)4-s + (−0.173 + 1.98i)5-s + (−3.14 − 2.61i)6-s + (2.42 − 4.19i)7-s + (−0.693 + 2.74i)8-s + (−1.82 − 5.02i)9-s + (2.78 + 0.477i)10-s + (−1.21 + 4.53i)11-s + (−4.04 + 4.12i)12-s + (−2.70 − 3.86i)13-s + (−5.62 − 3.90i)14-s + (4.41 + 3.70i)15-s + (3.78 + 1.29i)16-s + (0.240 + 0.0874i)17-s + ⋯ |
| L(s) = 1 | + (0.0824 − 0.996i)2-s + (0.956 − 1.36i)3-s + (−0.986 − 0.164i)4-s + (−0.0777 + 0.889i)5-s + (−1.28 − 1.06i)6-s + (0.915 − 1.58i)7-s + (−0.245 + 0.969i)8-s + (−0.609 − 1.67i)9-s + (0.879 + 0.150i)10-s + (−0.366 + 1.36i)11-s + (−1.16 + 1.19i)12-s + (−0.749 − 1.07i)13-s + (−1.50 − 1.04i)14-s + (1.14 + 0.957i)15-s + (0.945 + 0.324i)16-s + (0.0582 + 0.0212i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.448227 - 1.59992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.448227 - 1.59992i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.116 + 1.40i)T \) |
| 19 | \( 1 + (-3.06 - 3.10i)T \) |
| good | 3 | \( 1 + (-1.65 + 2.36i)T + (-1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (0.173 - 1.98i)T + (-4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-2.42 + 4.19i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.21 - 4.53i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (2.70 + 3.86i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.240 - 0.0874i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (0.364 + 0.305i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.47 - 3.15i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (0.365 - 0.632i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.69 - 4.69i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.869 - 4.93i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.48 - 0.217i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (2.37 + 6.52i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.0395 - 0.452i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-5.07 + 2.36i)T + (37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (0.912 + 10.4i)T + (-60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-6.29 - 2.93i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-0.999 - 1.19i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (9.36 - 1.65i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.359 - 2.03i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.76 - 6.60i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.298 - 1.69i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (4.00 - 10.9i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−11.38671175715191353365006425237, −10.39077837040472181706327946885, −9.787062304796239102872765107101, −8.120421225578823079019435543461, −7.64188310445768090920031929973, −6.92531513706905588926217065348, −4.95652365641868555148044481757, −3.56933609217710525773124900418, −2.47424345356784833358400710332, −1.26961272289594760902936447196,
2.73979179095711466294482412219, 4.25925818190491149989734795283, 5.05253738458136300294329388311, 5.75470903356795754355954645603, 7.72468110478092504609610177607, 8.630066704831610286671039316002, 8.967441986748144696573020295868, 9.607859356771341627597923269400, 11.15132288587086686620571864055, 12.13031628960254927045040919849
