L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (1.67 + 0.448i)7-s + (−0.707 + 0.707i)8-s + (−3 + 1.73i)11-s + (−3.34 + 0.896i)13-s + (0.866 − 1.50i)14-s + (0.500 + 0.866i)16-s + (−4.24 − 4.24i)17-s − 2i·19-s + (0.896 + 3.34i)22-s + (−0.776 − 2.89i)23-s + 3.46i·26-s + (−1.22 − 1.22i)28-s + (−4.33 − 7.5i)29-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.632 + 0.169i)7-s + (−0.249 + 0.249i)8-s + (−0.904 + 0.522i)11-s + (−0.928 + 0.248i)13-s + (0.231 − 0.400i)14-s + (0.125 + 0.216i)16-s + (−1.02 − 1.02i)17-s − 0.458i·19-s + (0.191 + 0.713i)22-s + (−0.161 − 0.604i)23-s + 0.679i·26-s + (−0.231 − 0.231i)28-s + (−0.804 − 1.39i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4478666983\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4478666983\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.67 - 0.448i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.34 - 0.896i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (4.24 + 4.24i)T + 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (0.776 + 2.89i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (4.33 + 7.5i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 + (7.5 + 4.33i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.68 + 10.0i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-0.776 + 2.89i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.13 + 11.7i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (-9.79 - 9.79i)T + 73iT^{2} \) |
| 79 | \( 1 + (3.46 - 2i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.89 + 0.776i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 + (6.69 + 1.79i)T + (84.0 + 48.5i)T^{2} \) |
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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
−9.230242713093552633705185391465, −8.553328631272937015025442565417, −7.49079885800345751037594564153, −6.85957962793414853951926126756, −5.39770426433729272124705562382, −4.92540948808272566610927460233, −4.03081496636584565603642656209, −2.59942965380390420528307941036, −2.04357741884569635898848372414, −0.15878361258769678038629898561,
1.83129739823083145923337201448, 3.17209078512230818504990629488, 4.28144050555374741973021944793, 5.12152658081859029983749723102, 5.85059879678669008301765648391, 6.79167106877334636607424053212, 7.896294690910265379138048583226, 8.002892319296761413220146053402, 9.151494969856481333185356989022, 9.925940017707459416129189171877