| L(s) = 1 | + (2.79 + 2.02i)2-s + (−1.49 + 2.60i)3-s + (2.44 + 7.52i)4-s + (1.27 − 4.83i)5-s + (−9.45 + 4.22i)6-s − 4.79i·7-s + (−4.17 + 12.8i)8-s + (−4.52 − 7.78i)9-s + (13.3 − 10.9i)10-s + (−3.78 + 5.20i)11-s + (−23.2 − 4.90i)12-s + (10.0 + 13.8i)13-s + (9.72 − 13.3i)14-s + (10.6 + 10.5i)15-s + (−12.1 + 8.79i)16-s + (7.53 − 23.1i)17-s + ⋯ |
| L(s) = 1 | + (1.39 + 1.01i)2-s + (−0.498 + 0.866i)3-s + (0.611 + 1.88i)4-s + (0.254 − 0.967i)5-s + (−1.57 + 0.704i)6-s − 0.684i·7-s + (−0.521 + 1.60i)8-s + (−0.502 − 0.864i)9-s + (1.33 − 1.09i)10-s + (−0.343 + 0.473i)11-s + (−1.93 − 0.408i)12-s + (0.771 + 1.06i)13-s + (0.694 − 0.956i)14-s + (0.711 + 0.702i)15-s + (−0.756 + 0.549i)16-s + (0.443 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44988 + 1.59369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44988 + 1.59369i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.49 - 2.60i)T \) |
| 5 | \( 1 + (-1.27 + 4.83i)T \) |
| good | 2 | \( 1 + (-2.79 - 2.02i)T + (1.23 + 3.80i)T^{2} \) |
| 7 | \( 1 + 4.79iT - 49T^{2} \) |
| 11 | \( 1 + (3.78 - 5.20i)T + (-37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-10.0 - 13.8i)T + (-52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-7.53 + 23.1i)T + (-233. - 169. i)T^{2} \) |
| 19 | \( 1 + (2.98 - 9.17i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (28.8 + 20.9i)T + (163. + 503. i)T^{2} \) |
| 29 | \( 1 + (11.9 - 3.89i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (16.0 - 49.3i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (16.7 + 23.0i)T + (-423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (9.29 + 12.7i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 8.20iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (1.74 + 5.37i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (0.677 + 2.08i)T + (-2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-46.4 - 63.8i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (13.8 + 10.0i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-13.6 - 4.43i)T + (3.63e3 + 2.63e3i)T^{2} \) |
| 71 | \( 1 + (18.4 - 5.98i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (76.5 - 105. i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (5.48 + 16.8i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-10.3 + 31.7i)T + (-5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (32.2 - 44.3i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-149. + 48.4i)T + (7.61e3 - 5.53e3i)T^{2} \) |
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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
−14.40914828802370249231087736033, −13.83631253361460246221668694967, −12.58039459535575717373370903760, −11.74763815647964213325998497660, −10.15143042499867663106011177620, −8.734643075066686826263404502145, −7.11278123652382572547354576381, −5.81192683359140410241956454936, −4.77583109122931512579476079463, −3.89854445042968361232463228382,
2.05190458304492342229197691406, 3.43731432546200614406467316919, 5.67449807352713905264779899397, 6.07467197393042975346604317762, 7.971379569818560341246611732764, 10.26913498297906658146617247733, 11.08380882424579130144257504596, 11.89843057808068420879220030591, 13.03348227886990000785286793508, 13.54326924680732910553920231864
