| 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
−17.70115003696419420009100125619, −16.27329286463341476755679851937, −15.52005546067142803602368437222, −13.94842229025436135185879828806, −12.69012674657279720579725209532, −11.22534612529670529243682962904, −9.556770051471529748751485762072, −8.701519957482883251445840148556, −5.66095007107135267912137327348, −4.73810456658168035860164939240,
3.98048819631383695223672152000, 6.54132635344714138363853998282, 7.73124663888318285965963720408, 9.836968353273208360514217343044, 11.31860272458010161865049727694, 12.73343442862007151362665344453, 13.69771012869302333356941173357, 14.92209439186877527469628575281, 16.93512229164178351082287067984, 17.70689602446413078878136299026
