L(s) = 1 | + (−0.580 + 1.40i)3-s + (−3.29 + 1.36i)5-s + (1.02 − 1.02i)7-s + (0.492 + 0.492i)9-s + (2.05 + 4.97i)11-s + (3.78 + 1.56i)13-s − 5.41i·15-s + 2.35i·17-s + (−2.79 − 1.15i)19-s + (0.843 + 2.03i)21-s + (−1.06 − 1.06i)23-s + (5.47 − 5.47i)25-s + (−5.18 + 2.14i)27-s + (1.60 − 3.86i)29-s − 10.5·31-s + ⋯ |
L(s) = 1 | + (−0.335 + 0.809i)3-s + (−1.47 + 0.610i)5-s + (0.387 − 0.387i)7-s + (0.164 + 0.164i)9-s + (0.620 + 1.49i)11-s + (1.04 + 0.434i)13-s − 1.39i·15-s + 0.570i·17-s + (−0.641 − 0.265i)19-s + (0.183 + 0.444i)21-s + (−0.221 − 0.221i)23-s + (1.09 − 1.09i)25-s + (−0.997 + 0.413i)27-s + (0.297 − 0.717i)29-s − 1.90·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7949305593\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7949305593\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.580 - 1.40i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (3.29 - 1.36i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.02 + 1.02i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.05 - 4.97i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-3.78 - 1.56i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 2.35iT - 17T^{2} \) |
| 19 | \( 1 + (2.79 + 1.15i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.06 + 1.06i)T + 23iT^{2} \) |
| 29 | \( 1 + (-1.60 + 3.86i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + (4.22 - 1.75i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-6.27 - 6.27i)T + 41iT^{2} \) |
| 43 | \( 1 + (3.56 + 8.60i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 3.06iT - 47T^{2} \) |
| 53 | \( 1 + (0.159 + 0.384i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (7.78 - 3.22i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.06 + 2.58i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (1.98 - 4.79i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-2.84 + 2.84i)T - 71iT^{2} \) |
| 73 | \( 1 + (8.43 + 8.43i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.59iT - 79T^{2} \) |
| 83 | \( 1 + (-7.60 - 3.14i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.967 + 0.967i)T - 89iT^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−10.71363161461686225553082560774, −9.635054145234122023146946906578, −8.704013663505732127003585553770, −7.69361815997348143447031623292, −7.16544088259299987472511897836, −6.20216898894428670590263113236, −4.70656417225516338398632947550, −4.19060248307836242186521149472, −3.61571552509103662490380535985, −1.80773245162903649241474507679,
0.40931716648352209947093104492, 1.46548427113500013714643873729, 3.40710918608940738566532511371, 4.00588542385459632593679784386, 5.33466568962599360895861302101, 6.14126892546847052492736463884, 7.10843313003751034183263448938, 7.933673764656188201313003104104, 8.587144642694966585935981911789, 9.139338547714631363530432387502