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
−11.21550393674169776082790600762, −10.37224549415620551594236521040, −9.427917730403299456012698832247, −7.69928927778539787370562120854, −7.35288796656819308006677545755, −6.15454107958198235756869119254, −5.23764469009509826808450669755, −4.34753554069383722890776656288, −3.60456289522428734898747984420, −2.46380710351974933734555144274,
1.76349000315946134722670635769, 2.94045975527092710862906226783, 3.67284530544948586589943290347, 5.14574006169213371146874178160, 5.92600313170482702987275645283, 6.45670384594682336186063618030, 7.82690135035638005808167525960, 8.833440745756746409271121131588, 10.32923281787953377791003895126, 10.93287488062177779578871148345