L(s) = 1 | + (−1.13 − 0.844i)2-s + (−1.79 + 1.03i)3-s + (0.572 + 1.91i)4-s + 5-s + (2.91 + 0.341i)6-s + (3.65 + 2.10i)7-s + (0.969 − 2.65i)8-s + (0.658 − 1.14i)9-s + (−1.13 − 0.844i)10-s + (−2.34 − 4.05i)11-s + (−3.02 − 2.85i)12-s + (3.60 + 0.189i)13-s + (−2.36 − 5.48i)14-s + (−1.79 + 1.03i)15-s + (−3.34 + 2.19i)16-s + (1.44 − 2.49i)17-s + ⋯ |
L(s) = 1 | + (−0.801 − 0.597i)2-s + (−1.03 + 0.599i)3-s + (0.286 + 0.958i)4-s + 0.447·5-s + (1.19 + 0.139i)6-s + (1.38 + 0.797i)7-s + (0.342 − 0.939i)8-s + (0.219 − 0.380i)9-s + (−0.358 − 0.267i)10-s + (−0.705 − 1.22i)11-s + (−0.872 − 0.823i)12-s + (0.998 + 0.0524i)13-s + (−0.631 − 1.46i)14-s + (−0.464 + 0.268i)15-s + (−0.836 + 0.548i)16-s + (0.349 − 0.605i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.796381 + 0.260381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.796381 + 0.260381i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.13 + 0.844i)T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-3.60 - 0.189i)T \) |
good | 3 | \( 1 + (1.79 - 1.03i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-3.65 - 2.10i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.34 + 4.05i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 2.49i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.09 - 5.36i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.36 - 5.82i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.64 + 2.68i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.540iT - 31T^{2} \) |
| 37 | \( 1 + (0.815 + 1.41i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.36 + 0.789i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.99 - 2.88i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 4.24iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 + (6.90 - 11.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.45 - 3.72i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.52 - 9.57i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.93 + 5.15i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 8.13iT - 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 - 3.34T + 83T^{2} \) |
| 89 | \( 1 + (2.76 - 1.59i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.2 + 7.09i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.86627605005663370706689251218, −10.45787369327991265725909781866, −9.241408818864349201915985013040, −8.429008603942203539794412099271, −7.74588590102519224213577279417, −6.01421454827691421676884965076, −5.51478778382515855011436974036, −4.29108469513602712416664195519, −2.78417040457083400144545837934, −1.28739766371890923452365242269,
0.882103951370454201762910237446, 2.00470521090412996144851018655, 4.66518865462716043984814554871, 5.26703748379984168530997338460, 6.51331477354060702325042643280, 6.98686164863124459060446955017, 8.026225366806255322250961459473, 8.769090280003424979757663586119, 10.12339430246743244909521323978, 10.86122329163413485650858103004