L(s) = 1 | + (2.53 − 0.679i)2-s + (0.258 − 0.965i)3-s + (4.23 − 2.44i)4-s − 2.62i·6-s + (−1.00 + 2.44i)7-s + (5.35 − 5.35i)8-s + (−0.866 − 0.499i)9-s + (−1.94 − 3.36i)11-s + (−1.26 − 4.71i)12-s + (0.707 + 0.707i)13-s + (−0.891 + 6.88i)14-s + (5.04 − 8.73i)16-s + (5.02 + 1.34i)17-s + (−2.53 − 0.679i)18-s + (−3.33 + 5.78i)19-s + ⋯ |
L(s) = 1 | + (1.79 − 0.480i)2-s + (0.149 − 0.557i)3-s + (2.11 − 1.22i)4-s − 1.07i·6-s + (−0.380 + 0.924i)7-s + (1.89 − 1.89i)8-s + (−0.288 − 0.166i)9-s + (−0.585 − 1.01i)11-s + (−0.364 − 1.36i)12-s + (0.196 + 0.196i)13-s + (−0.238 + 1.83i)14-s + (1.26 − 2.18i)16-s + (1.21 + 0.326i)17-s + (−0.597 − 0.160i)18-s + (−0.765 + 1.32i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.28987 - 2.18401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.28987 - 2.18401i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.00 - 2.44i)T \) |
good | 2 | \( 1 + (-2.53 + 0.679i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (1.94 + 3.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.02 - 1.34i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.33 - 5.78i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.66 + 6.20i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 7.76iT - 29T^{2} \) |
| 31 | \( 1 + (5.56 - 3.21i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0497 + 0.0133i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 5.55iT - 41T^{2} \) |
| 43 | \( 1 + (-4.97 + 4.97i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.547 - 2.04i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.01 - 0.540i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.61 + 6.26i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.396 - 1.47i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 + (-2.43 + 9.08i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.22 - 5.32i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.49 - 1.49i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7.46 + 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.34 - 3.34i)T - 97iT^{2} \) |
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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
−10.93287488062177779578871148345, −10.32923281787953377791003895126, −8.833440745756746409271121131588, −7.82690135035638005808167525960, −6.45670384594682336186063618030, −5.92600313170482702987275645283, −5.14574006169213371146874178160, −3.67284530544948586589943290347, −2.94045975527092710862906226783, −1.76349000315946134722670635769,
2.46380710351974933734555144274, 3.60456289522428734898747984420, 4.34753554069383722890776656288, 5.23764469009509826808450669755, 6.15454107958198235756869119254, 7.35288796656819308006677545755, 7.69928927778539787370562120854, 9.427917730403299456012698832247, 10.37224549415620551594236521040, 11.21550393674169776082790600762