| L(s) = 1 | + (1.34 − 0.426i)2-s + (−0.730 − 0.951i)3-s + (1.63 − 1.14i)4-s + (1.76 + 1.35i)5-s + (−1.39 − 0.971i)6-s + (1.00 + 2.44i)7-s + (1.71 − 2.24i)8-s + (0.404 − 1.50i)9-s + (2.95 + 1.07i)10-s + (−5.85 − 0.770i)11-s + (−2.28 − 0.717i)12-s + (0.925 + 0.383i)13-s + (2.39 + 2.87i)14-s − 2.66i·15-s + (1.35 − 3.76i)16-s + (−1.66 − 0.962i)17-s + ⋯ |
| L(s) = 1 | + (0.953 − 0.301i)2-s + (−0.421 − 0.549i)3-s + (0.818 − 0.574i)4-s + (0.788 + 0.605i)5-s + (−0.567 − 0.396i)6-s + (0.379 + 0.925i)7-s + (0.606 − 0.794i)8-s + (0.134 − 0.502i)9-s + (0.934 + 0.339i)10-s + (−1.76 − 0.232i)11-s + (−0.660 − 0.207i)12-s + (0.256 + 0.106i)13-s + (0.640 + 0.767i)14-s − 0.688i·15-s + (0.339 − 0.940i)16-s + (−0.404 − 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89340 - 0.686629i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89340 - 0.686629i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.34 + 0.426i)T \) |
| 7 | \( 1 + (-1.00 - 2.44i)T \) |
| good | 3 | \( 1 + (0.730 + 0.951i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (-1.76 - 1.35i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (5.85 + 0.770i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (-0.925 - 0.383i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (1.66 + 0.962i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.467 - 3.55i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-0.613 + 2.29i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.25 - 5.45i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (2.43 - 4.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.355 + 0.273i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (-6.56 - 6.56i)T + 41iT^{2} \) |
| 43 | \( 1 + (-0.154 - 0.372i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (1.76 - 1.01i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.01 + 0.265i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-1.81 + 13.8i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (10.4 - 1.38i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-4.46 - 5.82i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-11.4 + 11.4i)T - 71iT^{2} \) |
| 73 | \( 1 + (-13.4 + 3.60i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.15 + 3.55i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (12.5 + 5.21i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (2.76 + 0.739i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 1.30T + 97T^{2} \) |
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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
−12.45525032113494100333665514613, −11.24466206746626313748125348405, −10.61024681356682080512280364735, −9.486819012653996379847959884366, −7.910346226424852328535674589536, −6.64255580142847218103208514329, −5.86613943246050242271518597094, −5.04734319113124708273177511317, −3.09867986986822595278777921897, −1.96219678521126597197888214624,
2.25484345980955978826138453798, 4.14975970394749246834800360592, 5.09328012301628391223161278372, 5.69338983820504880886849224803, 7.28492306187823686716311410620, 8.081246236995454588321670608255, 9.699629753188037306345956799801, 10.74404430475221496761683312367, 11.18634396694464693591398180326, 12.76813786864523102473127662315
