L(s) = 1 | + (1.30 − 1.14i)3-s + (−0.761 − 0.639i)5-s + (1.35 + 0.492i)7-s + (0.396 − 2.97i)9-s + (−2.56 + 2.15i)11-s + (−0.337 + 1.91i)13-s + (−1.72 + 0.0362i)15-s + (1.16 − 2.01i)17-s + (3.38 + 5.86i)19-s + (2.32 − 0.901i)21-s + (−8.41 + 3.06i)23-s + (−0.696 − 3.95i)25-s + (−2.87 − 4.32i)27-s + (0.847 + 4.80i)29-s + (−5.81 + 2.11i)31-s + ⋯ |
L(s) = 1 | + (0.752 − 0.658i)3-s + (−0.340 − 0.285i)5-s + (0.511 + 0.186i)7-s + (0.132 − 0.991i)9-s + (−0.773 + 0.648i)11-s + (−0.0936 + 0.531i)13-s + (−0.444 + 0.00936i)15-s + (0.281 − 0.487i)17-s + (0.776 + 1.34i)19-s + (0.507 − 0.196i)21-s + (−1.75 + 0.638i)23-s + (−0.139 − 0.790i)25-s + (−0.553 − 0.832i)27-s + (0.157 + 0.892i)29-s + (−1.04 + 0.380i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16294 - 0.344978i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16294 - 0.344978i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.30 + 1.14i)T \) |
good | 5 | \( 1 + (0.761 + 0.639i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.35 - 0.492i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (2.56 - 2.15i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.337 - 1.91i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.16 + 2.01i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.38 - 5.86i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (8.41 - 3.06i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.847 - 4.80i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.81 - 2.11i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (0.0829 - 0.143i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.87 + 10.6i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.82 + 5.73i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.14 - 2.23i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + (-3.02 - 2.53i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.91 + 2.88i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.00383 - 0.0217i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.53 + 7.85i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.26 + 3.92i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.00 + 5.70i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.898 - 5.09i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-1.91 - 3.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.40 - 4.53i)T + (16.8 - 95.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.83258901621926210898624899569, −12.33253453010258350738249318867, −12.04402383381191244047776887125, −10.31404919118047683982170828863, −9.140105418181650584314450461679, −7.983423096356040017579562531040, −7.30976276931837044449447368007, −5.57317879583325711418212384971, −3.87948875983784716088081480211, −2.01684233813988193756287545892,
2.75974797778797820020452839801, 4.17572146636358562765252027995, 5.58760639510411974810927286354, 7.57468562286328620561759570510, 8.240226442706818056956493053688, 9.568085123575681398692937804473, 10.63109700699153775137892383425, 11.43929052171744359803271737782, 13.03294517667520252198758855801, 13.91316647814853541167106279489