| L(s) = 1 | + (0.174 − 0.0566i)2-s + (−0.865 − 1.19i)3-s + (−1.59 + 1.15i)4-s + (0.107 + 2.23i)5-s + (−0.218 − 0.158i)6-s − 3.26i·7-s + (−0.427 + 0.587i)8-s + (0.257 − 0.792i)9-s + (0.145 + 0.382i)10-s + (0.618 + 1.90i)11-s + (2.75 + 0.894i)12-s + (0.281 + 0.0915i)13-s + (−0.184 − 0.568i)14-s + (2.56 − 2.06i)15-s + (1.17 − 3.61i)16-s + (−3.03 + 4.17i)17-s + ⋯ |
| L(s) = 1 | + (0.123 − 0.0400i)2-s + (−0.499 − 0.687i)3-s + (−0.795 + 0.577i)4-s + (0.0481 + 0.998i)5-s + (−0.0890 − 0.0646i)6-s − 1.23i·7-s + (−0.150 + 0.207i)8-s + (0.0858 − 0.264i)9-s + (0.0459 + 0.121i)10-s + (0.186 + 0.573i)11-s + (0.794 + 0.258i)12-s + (0.0781 + 0.0254i)13-s + (−0.0493 − 0.151i)14-s + (0.662 − 0.532i)15-s + (0.293 − 0.903i)16-s + (−0.736 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.562824 - 0.0437038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.562824 - 0.0437038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.107 - 2.23i)T \) |
| good | 2 | \( 1 + (-0.174 + 0.0566i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.865 + 1.19i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 3.26iT - 7T^{2} \) |
| 11 | \( 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.281 - 0.0915i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (3.03 - 4.17i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.39 - 1.01i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.836 + 0.271i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.78 + 3.47i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.93 + 3.58i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-7.69 - 2.49i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.313 - 0.965i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.24iT - 43T^{2} \) |
| 47 | \( 1 + (2.48 + 3.41i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (4.76 + 6.55i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.83 - 5.64i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.282 - 0.870i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-4.04 + 5.57i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-4.82 + 3.50i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.40 + 2.72i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (6.27 - 4.56i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (8.53 - 11.7i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (2.32 + 7.15i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.95 - 5.44i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−17.70689602446413078878136299026, −16.93512229164178351082287067984, −14.92209439186877527469628575281, −13.69771012869302333356941173357, −12.73343442862007151362665344453, −11.31860272458010161865049727694, −9.836968353273208360514217343044, −7.73124663888318285965963720408, −6.54132635344714138363853998282, −3.98048819631383695223672152000,
4.73810456658168035860164939240, 5.66095007107135267912137327348, 8.701519957482883251445840148556, 9.556770051471529748751485762072, 11.22534612529670529243682962904, 12.69012674657279720579725209532, 13.94842229025436135185879828806, 15.52005546067142803602368437222, 16.27329286463341476755679851937, 17.70115003696419420009100125619
