L(s) = 1 | + (−1.25 − 1.19i)3-s + (0.846 + 1.46i)5-s + (0.5 − 0.866i)7-s + (0.150 + 2.99i)9-s + (0.474 − 0.822i)11-s + (1.69 + 2.93i)13-s + (0.687 − 2.84i)15-s − 0.300·17-s + 5.73·19-s + (−1.66 + 0.490i)21-s + (1.17 + 2.03i)23-s + (1.06 − 1.84i)25-s + (3.38 − 3.93i)27-s + (1.01 − 1.76i)29-s + (−0.522 − 0.905i)31-s + ⋯ |
L(s) = 1 | + (−0.724 − 0.689i)3-s + (0.378 + 0.655i)5-s + (0.188 − 0.327i)7-s + (0.0501 + 0.998i)9-s + (0.143 − 0.247i)11-s + (0.470 + 0.814i)13-s + (0.177 − 0.735i)15-s − 0.0729·17-s + 1.31·19-s + (−0.362 + 0.106i)21-s + (0.244 + 0.423i)23-s + (0.213 − 0.369i)25-s + (0.651 − 0.758i)27-s + (0.189 − 0.327i)29-s + (−0.0938 − 0.162i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27790 - 0.0795873i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27790 - 0.0795873i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.25 + 1.19i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.846 - 1.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.474 + 0.822i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.69 - 2.93i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 0.300T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 + (-1.17 - 2.03i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.01 + 1.76i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.522 + 0.905i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.22T + 37T^{2} \) |
| 41 | \( 1 + (-2.56 - 4.44i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.25 + 9.10i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.24 - 5.61i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + (-4.46 - 7.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.256 - 0.443i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.63 + 11.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 + 1.59T + 73T^{2} \) |
| 79 | \( 1 + (6.43 - 11.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.98 - 3.43i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6.24T + 89T^{2} \) |
| 97 | \( 1 + (5.02 - 8.70i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.10414812193419237279329368890, −10.19589809981912776433903978028, −9.211161389096922358555321722284, −7.949098720501939912292145447882, −7.12708609250943956821434350928, −6.34496466847412255271708428995, −5.50664539372799846990820960655, −4.23513545567650018357849503900, −2.68688176556172322915324669396, −1.23986160624913363632404373819,
1.10728347527525088215319670116, 3.09839677271246997707148280244, 4.43618074622070580946554376186, 5.33473907118710790653890705334, 5.96354462998806767227984092196, 7.22883677843161540482047651144, 8.479620057748325065479698849510, 9.307499505107241141084263341978, 10.00110824401446811618420506996, 10.96966935728231343141367653783