| L(s) = 1 | + (7.48 − 1.32i)3-s + (−18.9 + 15.8i)5-s + (−39.3 + 68.2i)7-s + (−21.7 + 7.93i)9-s + (20.6 + 35.7i)11-s + (132. + 23.4i)13-s + (−120. + 143. i)15-s + (120. + 43.7i)17-s + (−321. − 163. i)19-s + (−204. + 562. i)21-s + (360. + 302. i)23-s + (−2.69 + 15.2i)25-s + (−686. + 396. i)27-s + (15.3 + 42.2i)29-s + (469. + 271. i)31-s + ⋯ |
| L(s) = 1 | + (0.831 − 0.146i)3-s + (−0.756 + 0.634i)5-s + (−0.803 + 1.39i)7-s + (−0.269 + 0.0979i)9-s + (0.170 + 0.295i)11-s + (0.786 + 0.138i)13-s + (−0.536 + 0.639i)15-s + (0.415 + 0.151i)17-s + (−0.891 − 0.452i)19-s + (−0.464 + 1.27i)21-s + (0.680 + 0.571i)23-s + (−0.00430 + 0.0244i)25-s + (−0.941 + 0.543i)27-s + (0.0182 + 0.0502i)29-s + (0.488 + 0.282i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.823877 + 1.09732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.823877 + 1.09732i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (321. + 163. i)T \) |
| good | 3 | \( 1 + (-7.48 + 1.32i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (18.9 - 15.8i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (39.3 - 68.2i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-20.6 - 35.7i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-132. - 23.4i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-120. - 43.7i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-360. - 302. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-15.3 - 42.2i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-469. - 271. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.45e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.48e3 - 261. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-2.13e3 + 1.79e3i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-3.36e3 + 1.22e3i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (3.04e3 - 3.63e3i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (165. - 455. i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-4.39e3 - 3.68e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (1.80e3 + 4.95e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-4.21e3 - 5.02e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-746. - 4.23e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (2.53e3 - 446. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-424. + 735. i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (2.58e3 + 455. i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (1.83e3 - 5.03e3i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−14.21679131458182294295950083803, −13.00827437526802005214928713546, −11.96798464102090850889898237653, −10.86983836973494445880011042531, −9.233605282116168948839925251812, −8.523438814002885876803609965928, −7.16752298465344490503205485632, −5.79314444051148093180496852635, −3.60000693648282947579467407044, −2.49239094492401657839700711059,
0.63910691977510124871607872174, 3.30557056828772739139164111843, 4.25988853896941232583893023958, 6.40145660379682040578936033782, 7.87418304534010420736638326659, 8.708128268507220932352991100622, 9.978444419077045900504334090811, 11.16945270530497616591856492618, 12.57004549160529187307498479744, 13.52306712768298449436331177971
