| L(s) = 1 | + (−2.14 + 0.695i)2-s + (1.08 + 1.48i)3-s + (2.48 − 1.80i)4-s + (−0.133 + 2.23i)5-s + (−3.35 − 2.43i)6-s − 4.38i·7-s + (−1.41 + 1.94i)8-s + (−0.119 + 0.366i)9-s + (−1.26 − 4.87i)10-s + (1.54 + 4.76i)11-s + (5.37 + 1.74i)12-s + (5.16 + 1.67i)13-s + (3.04 + 9.38i)14-s + (−3.46 + 2.21i)15-s + (−0.222 + 0.684i)16-s + (0.587 − 0.809i)17-s + ⋯ |
| L(s) = 1 | + (−1.51 + 0.491i)2-s + (0.624 + 0.859i)3-s + (1.24 − 0.901i)4-s + (−0.0596 + 0.998i)5-s + (−1.36 − 0.994i)6-s − 1.65i·7-s + (−0.500 + 0.688i)8-s + (−0.0396 + 0.122i)9-s + (−0.400 − 1.54i)10-s + (0.466 + 1.43i)11-s + (1.55 + 0.503i)12-s + (1.43 + 0.465i)13-s + (0.814 + 2.50i)14-s + (−0.895 + 0.572i)15-s + (−0.0555 + 0.171i)16-s + (0.142 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.528538 + 0.685894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.528538 + 0.685894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.133 - 2.23i)T \) |
| 17 | \( 1 + (-0.587 + 0.809i)T \) |
| good | 2 | \( 1 + (2.14 - 0.695i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.08 - 1.48i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 4.38iT - 7T^{2} \) |
| 11 | \( 1 + (-1.54 - 4.76i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-5.16 - 1.67i)T + (10.5 + 7.64i)T^{2} \) |
| 19 | \( 1 + (-2.84 - 2.06i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.72 - 1.21i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (1.35 - 0.985i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.45 - 3.23i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (5.69 + 1.85i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.38 - 4.25i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.73iT - 43T^{2} \) |
| 47 | \( 1 + (2.75 + 3.79i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.49 - 6.18i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.96 - 6.03i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.481 + 1.48i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (6.97 - 9.59i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-12.2 + 8.88i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.39 + 0.452i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (3.22 - 2.34i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.78 + 9.33i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.22 + 12.9i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.46 - 8.89i)T + (-29.9 + 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76782632321298363096794426854, −10.18604477718019550643695960200, −9.788240159991653714943732818402, −8.841462282231733962646189552326, −7.76221765750108412784067173346, −7.08315482856117054120978512018, −6.41458488992511943838146905084, −4.23132750241867276764706914855, −3.55831345901174160657158449827, −1.49247562592583273868469677623,
0.984768289852973150403920779671, 2.06437984261658816516375056436, 3.26173519025932452898109678884, 5.41114428140109611877282999692, 6.37525518322675466050321427774, 8.001881148333532444629649094165, 8.381696369745513375390588191999, 8.809785495025710811215008230631, 9.609348944499578900266284547483, 10.96796243120317824706704437785
