L(s) = 1 | − 1.70i·2-s − 0.922·4-s + (1.84 − 1.06i)5-s + (1.43 − 2.22i)7-s − 1.84i·8-s + (−1.81 − 3.15i)10-s + (2.01 − 1.16i)11-s + (−2.64 + 2.44i)13-s + (−3.79 − 2.45i)14-s − 4.99·16-s + 0.207·17-s + (2.00 + 1.16i)19-s + (−1.70 + 0.981i)20-s + (−1.98 − 3.44i)22-s + 5.92·23-s + ⋯ |
L(s) = 1 | − 1.20i·2-s − 0.461·4-s + (0.824 − 0.475i)5-s + (0.543 − 0.839i)7-s − 0.651i·8-s + (−0.575 − 0.996i)10-s + (0.607 − 0.350i)11-s + (−0.733 + 0.679i)13-s + (−1.01 − 0.656i)14-s − 1.24·16-s + 0.0504·17-s + (0.460 + 0.266i)19-s + (−0.380 + 0.219i)20-s + (−0.423 − 0.734i)22-s + 1.23·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.632563 - 1.89132i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.632563 - 1.89132i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.43 + 2.22i)T \) |
| 13 | \( 1 + (2.64 - 2.44i)T \) |
good | 2 | \( 1 + 1.70iT - 2T^{2} \) |
| 5 | \( 1 + (-1.84 + 1.06i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.01 + 1.16i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 0.207T + 17T^{2} \) |
| 19 | \( 1 + (-2.00 - 1.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.92T + 23T^{2} \) |
| 29 | \( 1 + (-2.10 + 3.65i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.50 + 2.02i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.74iT - 37T^{2} \) |
| 41 | \( 1 + (-2.48 - 1.43i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.38 + 5.85i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (10.6 - 6.13i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.04 + 3.53i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 1.29iT - 59T^{2} \) |
| 61 | \( 1 + (0.470 - 0.815i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.52 - 2.03i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.10 - 0.638i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.60 - 2.66i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.77 - 9.99i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.31iT - 83T^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 + (-3.07 + 1.77i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.765284709373637862599631519150, −9.592589680524606651272299614048, −8.467490860520502879972618179070, −7.25749014251476695523898077446, −6.47120135133461900936227023667, −5.14753165514566227505852072871, −4.29036944049743720416516499782, −3.21342748395573693159317826562, −1.91331031465306938528668341603, −1.07117613470803242609312838502,
1.92827256141307301899054326874, 2.96796853575523556825359053045, 4.84726660532055847974989483417, 5.41426265050825649802146945698, 6.27740143394874543823965828393, 7.05825093022628947579755062757, 7.80211485377021891584426616996, 8.833895980878437664839912560018, 9.417975540624856580691414612234, 10.50266036359513309667883680749