L(s) = 1 | + (−0.422 − 0.906i)2-s + (−1.61 − 0.629i)3-s + (−0.642 + 0.766i)4-s + (−1.65 − 1.49i)5-s + (0.111 + 1.72i)6-s + (0.539 + 2.01i)7-s + (0.965 + 0.258i)8-s + (2.20 + 2.03i)9-s + (−0.657 + 2.13i)10-s + (−3.78 + 2.18i)11-s + (1.51 − 0.831i)12-s + (3.22 − 2.25i)13-s + (1.59 − 1.33i)14-s + (1.73 + 3.46i)15-s + (−0.173 − 0.984i)16-s + (2.50 + 5.38i)17-s + ⋯ |
L(s) = 1 | + (−0.298 − 0.640i)2-s + (−0.931 − 0.363i)3-s + (−0.321 + 0.383i)4-s + (−0.741 − 0.670i)5-s + (0.0454 + 0.705i)6-s + (0.203 + 0.760i)7-s + (0.341 + 0.0915i)8-s + (0.735 + 0.677i)9-s + (−0.207 + 0.675i)10-s + (−1.14 + 0.659i)11-s + (0.438 − 0.239i)12-s + (0.894 − 0.626i)13-s + (0.426 − 0.358i)14-s + (0.447 + 0.894i)15-s + (−0.0434 − 0.246i)16-s + (0.608 + 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.685420 - 0.208473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.685420 - 0.208473i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.422 + 0.906i)T \) |
| 3 | \( 1 + (1.61 + 0.629i)T \) |
| 5 | \( 1 + (1.65 + 1.49i)T \) |
| 19 | \( 1 + (3.26 + 2.88i)T \) |
good | 7 | \( 1 + (-0.539 - 2.01i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (3.78 - 2.18i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.22 + 2.25i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-2.50 - 5.38i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-5.77 + 0.504i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (-3.16 + 1.15i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.67 + 4.63i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.42 + 1.42i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.40 - 0.424i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.38 - 0.208i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-2.85 - 1.33i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-12.3 + 1.08i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (4.57 + 1.66i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.07 - 5.09i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (6.48 - 13.9i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-7.74 - 9.22i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.89 + 4.12i)T + (-24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (2.39 - 0.421i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.621 - 2.31i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (1.17 - 6.64i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 4.80i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.77452541210348731247599610454, −10.10309407524385419047850767536, −8.710954810178844158440224133500, −8.180053207938286078970629566401, −7.25447220827740433317292954190, −5.86321313320180184722013954095, −5.05915734064662371298233483751, −4.06950222095477120924685083552, −2.44300359820318226527007371247, −0.941022071518293648774373375522,
0.73642084100324430828372888357, 3.27961058055456471263576441282, 4.40241810346137927350214674423, 5.32953737066130880948298723173, 6.43781786321621533168837327592, 7.13969649132568622770740353663, 7.964651778856381030160579808240, 8.992872523870493822233503924044, 10.29118987383595408824468310965, 10.68451097894842717706819390599