L(s) = 1 | + (0.445 − 0.184i)3-s + (−0.565 + 1.36i)5-s + (−0.135 − 0.135i)7-s + (−1.95 + 1.95i)9-s + (3.12 + 1.29i)11-s + (−0.951 − 2.29i)13-s + 0.713i·15-s − 3.11i·17-s + (2.48 + 5.99i)19-s + (−0.0851 − 0.0352i)21-s + (−5.18 + 5.18i)23-s + (1.98 + 1.98i)25-s + (−1.06 + 2.57i)27-s + (4.33 − 1.79i)29-s − 7.44·31-s + ⋯ |
L(s) = 1 | + (0.257 − 0.106i)3-s + (−0.253 + 0.610i)5-s + (−0.0510 − 0.0510i)7-s + (−0.652 + 0.652i)9-s + (0.941 + 0.390i)11-s + (−0.263 − 0.637i)13-s + 0.184i·15-s − 0.754i·17-s + (0.569 + 1.37i)19-s + (−0.0185 − 0.00769i)21-s + (−1.08 + 1.08i)23-s + (0.397 + 0.397i)25-s + (−0.204 + 0.494i)27-s + (0.804 − 0.333i)29-s − 1.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.315665349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.315665349\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.445 + 0.184i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.565 - 1.36i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (0.135 + 0.135i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.12 - 1.29i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (0.951 + 2.29i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 3.11iT - 17T^{2} \) |
| 19 | \( 1 + (-2.48 - 5.99i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (5.18 - 5.18i)T - 23iT^{2} \) |
| 29 | \( 1 + (-4.33 + 1.79i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 + (3.49 - 8.44i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (4.27 - 4.27i)T - 41iT^{2} \) |
| 43 | \( 1 + (-4.33 - 1.79i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 12.0iT - 47T^{2} \) |
| 53 | \( 1 + (3.42 + 1.41i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (1.16 - 2.81i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-8.72 + 3.61i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-7.15 + 2.96i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-2.86 - 2.86i)T + 71iT^{2} \) |
| 73 | \( 1 + (2.49 - 2.49i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (-5.42 - 13.0i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (4.96 + 4.96i)T + 89iT^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−10.02277452621638742543004072539, −9.497825465055870636928648562803, −8.312219888704555162770982424882, −7.71058852875189208239463146862, −6.93235600983894995831680420059, −5.89419953499031804358905943358, −5.02130125332495961563545351679, −3.70968341170567722479832777644, −2.95472261041981062661033568085, −1.63132172656460376261403455734,
0.58303933872093493186957084339, 2.21004339098635117713790352841, 3.54492528051049051203802279985, 4.28260863183781541780819126925, 5.39318124012587362572411450933, 6.39099980593589922640638781713, 7.11322965227805819015537589324, 8.446332428138851545656629504647, 8.827664010392388813847949876245, 9.449154233954235453482391282788