| L(s) = 1 | + (−3.03 + 3.03i)2-s + (−7.54 − 7.54i)3-s − 2.46i·4-s + 45.8·6-s + (−13.0 + 13.0i)7-s + (−41.1 − 41.1i)8-s + 32.8i·9-s + 5.95·11-s + (−18.6 + 18.6i)12-s + (−143. − 143. i)13-s − 79.5i·14-s + 289.·16-s + (169. − 169. i)17-s + (−99.8 − 99.8i)18-s − 160. i·19-s + ⋯ |
| L(s) = 1 | + (−0.759 + 0.759i)2-s + (−0.838 − 0.838i)3-s − 0.154i·4-s + 1.27·6-s + (−0.267 + 0.267i)7-s + (−0.642 − 0.642i)8-s + 0.405i·9-s + 0.0492·11-s + (−0.129 + 0.129i)12-s + (−0.849 − 0.849i)13-s − 0.406i·14-s + 1.13·16-s + (0.586 − 0.586i)17-s + (−0.308 − 0.308i)18-s − 0.445i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4319519571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4319519571\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + (13.0 - 13.0i)T \) |
| good | 2 | \( 1 + (3.03 - 3.03i)T - 16iT^{2} \) |
| 3 | \( 1 + (7.54 + 7.54i)T + 81iT^{2} \) |
| 11 | \( 1 - 5.95T + 1.46e4T^{2} \) |
| 13 | \( 1 + (143. + 143. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-169. + 169. i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 160. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (144. + 144. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 - 1.47e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 511.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (352. - 352. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.33e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-2.29e3 - 2.29e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (2.34e3 - 2.34e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.76e3 - 1.76e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + 3.72e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 150.T + 1.38e7T^{2} \) |
| 67 | \( 1 + (2.55e3 - 2.55e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 1.51e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-6.85e3 - 6.85e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 2.16e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-3.03e3 - 3.03e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 8.46e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-7.88e3 + 7.88e3i)T - 8.85e7iT^{2} \) |
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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
−12.47492742988626806284356214360, −11.46352415974909859800373270734, −10.10266056597341055881364408536, −9.126691916030555987121731900221, −7.925021376136970093376382802712, −7.11503370928891536042199727837, −6.29959795707841973024073634004, −5.20202560022795719592085787703, −3.04738108523830645910365981012, −0.875685693531344907000308718882,
0.31795836775266688927810076118, 2.07508488379838334871467585598, 3.89079439716980245353821272194, 5.19950451941765039154872444684, 6.25791139347189479852952282800, 7.86741717915804801091785159837, 9.216548072721635450752196475647, 10.06876184151922646101230630200, 10.47379971318125518475392863447, 11.65845873686783316625128077521
