L(s) = 1 | + (0.545 − 1.92i)2-s + (3.88 − 2.24i)3-s + (−3.40 − 2.09i)4-s + (−0.133 − 0.231i)5-s + (−2.19 − 8.70i)6-s + 7.24i·7-s + (−5.89 + 5.40i)8-s + (5.56 − 9.64i)9-s + (−0.518 + 0.130i)10-s + 11.7i·11-s + (−17.9 − 0.519i)12-s + (4.17 − 7.22i)13-s + (13.9 + 3.95i)14-s + (−1.03 − 0.600i)15-s + (7.18 + 14.2i)16-s + (−7.11 − 12.3i)17-s + ⋯ |
L(s) = 1 | + (0.272 − 0.962i)2-s + (1.29 − 0.747i)3-s + (−0.851 − 0.524i)4-s + (−0.0267 − 0.0463i)5-s + (−0.366 − 1.45i)6-s + 1.03i·7-s + (−0.737 + 0.675i)8-s + (0.618 − 1.07i)9-s + (−0.0518 + 0.0130i)10-s + 1.06i·11-s + (−1.49 − 0.0433i)12-s + (0.320 − 0.555i)13-s + (0.996 + 0.282i)14-s + (−0.0693 − 0.0400i)15-s + (0.449 + 0.893i)16-s + (−0.418 − 0.725i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0843 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0843 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.23330 - 1.34214i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23330 - 1.34214i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.545 + 1.92i)T \) |
| 19 | \( 1 + (11.9 + 14.7i)T \) |
good | 3 | \( 1 + (-3.88 + 2.24i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (0.133 + 0.231i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 - 7.24iT - 49T^{2} \) |
| 11 | \( 1 - 11.7iT - 121T^{2} \) |
| 13 | \( 1 + (-4.17 + 7.22i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (7.11 + 12.3i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-21.4 - 12.4i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (25.7 - 44.6i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 26.3iT - 961T^{2} \) |
| 37 | \( 1 + 20.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + (38.5 + 66.8i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-30.0 + 17.3i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (31.9 + 18.4i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (8.75 - 15.1i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (82.8 - 47.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-26.2 + 45.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (11.8 + 6.86i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (39.7 - 22.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-20.0 - 34.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-106. + 61.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 108. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (26.2 - 45.4i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-29.9 - 51.7i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−13.76289707486164334205526210323, −12.81386019473007339600902575352, −12.21390698074816697036938346230, −10.71136176494295565688703647292, −9.139766316233918896800355710839, −8.707760579324136569571476556836, −7.07839011160565786317392193050, −5.06926069102626274122332171599, −3.12417563612422609833126016120, −1.99090073595735865248797190134,
3.45928122962962461255119060161, 4.36209441630565040641674475564, 6.31724588793799666508238805748, 7.82010378758799671488570804180, 8.650256729171675958194698056696, 9.685851268514759748087500670653, 11.01999803336501113395198126558, 13.11234958920264904441667487657, 13.72791740166631763653323994352, 14.65743788968590825463526240359