L(s) = 1 | + (0.707 − 0.707i)2-s + (1.42 − 1.42i)3-s − 1.00i·4-s + (0.204 − 2.22i)5-s − 2.02i·6-s + (−0.707 − 0.707i)8-s − 1.08i·9-s + (−1.42 − 1.71i)10-s − 4.03·11-s + (−1.42 − 1.42i)12-s + (0.204 − 0.204i)13-s + (−2.89 − 3.47i)15-s − 1.00·16-s + (1.44 + 1.44i)17-s + (−0.769 − 0.769i)18-s + 6.20·19-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.825 − 0.825i)3-s − 0.500i·4-s + (0.0916 − 0.995i)5-s − 0.825i·6-s + (−0.250 − 0.250i)8-s − 0.362i·9-s + (−0.452 − 0.543i)10-s − 1.21·11-s + (−0.412 − 0.412i)12-s + (0.0568 − 0.0568i)13-s + (−0.746 − 0.897i)15-s − 0.250·16-s + (0.349 + 0.349i)17-s + (−0.181 − 0.181i)18-s + 1.42·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09374 - 2.00889i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09374 - 2.00889i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.204 + 2.22i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.42 + 1.42i)T - 3iT^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 + (-0.204 + 0.204i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.44 - 1.44i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 + (-3.20 - 3.20i)T + 23iT^{2} \) |
| 29 | \( 1 - 7.15iT - 29T^{2} \) |
| 31 | \( 1 + 7.31iT - 31T^{2} \) |
| 37 | \( 1 + (-3.27 + 3.27i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.58iT - 41T^{2} \) |
| 43 | \( 1 + (4.97 + 4.97i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.222 + 0.222i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.85 - 5.85i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.855T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + (2.23 - 2.23i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 + (8.15 - 8.15i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.07iT - 79T^{2} \) |
| 83 | \( 1 + (3.85 - 3.85i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.07T + 89T^{2} \) |
| 97 | \( 1 + (6.63 + 6.63i)T + 97iT^{2} \) |
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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
−10.73287177496011890494250351374, −9.697561716042693909641085313449, −8.847700872898961051533853511664, −7.915907207882145924844606424933, −7.25844358564364787798009516291, −5.64148136587918857402891622833, −5.01949899403594998225702658153, −3.53059743841958877120603826015, −2.41935403177018642446999542105, −1.20603263692282532759071027584,
2.76432225100279497959633066689, 3.24391210412703905596008485411, 4.53481062947817456681044568427, 5.54459176729655763351466473271, 6.72128607918530071957573171037, 7.65034206424659176125337277536, 8.433239406596272078829423207949, 9.632048901853979394809016663570, 10.16854819423309212234967147876, 11.17832380031520018781297370492