| L(s) = 1 | + (0.253 − 1.98i)2-s + (1.04 − 3.90i)3-s + (−3.87 − 1.00i)4-s + (4.34 − 2.48i)5-s + (−7.48 − 3.06i)6-s + (6.56 − 2.42i)7-s + (−2.97 + 7.42i)8-s + (−6.36 − 3.67i)9-s + (−3.81 − 9.24i)10-s + (4.38 − 2.52i)11-s + (−7.98 + 14.0i)12-s + (−14.0 + 14.0i)13-s + (−3.14 − 13.6i)14-s + (−5.14 − 19.5i)15-s + (13.9 + 7.79i)16-s + (−5.60 + 20.9i)17-s + ⋯ |
| L(s) = 1 | + (0.126 − 0.991i)2-s + (0.348 − 1.30i)3-s + (−0.967 − 0.251i)4-s + (0.868 − 0.496i)5-s + (−1.24 − 0.511i)6-s + (0.938 − 0.346i)7-s + (−0.372 + 0.928i)8-s + (−0.706 − 0.408i)9-s + (−0.381 − 0.924i)10-s + (0.398 − 0.229i)11-s + (−0.665 + 1.17i)12-s + (−1.08 + 1.08i)13-s + (−0.224 − 0.974i)14-s + (−0.342 − 1.30i)15-s + (0.873 + 0.486i)16-s + (−0.329 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.415472 - 1.76707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.415472 - 1.76707i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.253 + 1.98i)T \) |
| 5 | \( 1 + (-4.34 + 2.48i)T \) |
| 7 | \( 1 + (-6.56 + 2.42i)T \) |
| good | 3 | \( 1 + (-1.04 + 3.90i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (-4.38 + 2.52i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (14.0 - 14.0i)T - 169iT^{2} \) |
| 17 | \( 1 + (5.60 - 20.9i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-6.74 - 3.89i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-3.35 - 12.5i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 17.3iT - 841T^{2} \) |
| 31 | \( 1 + (16.9 + 29.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (0.822 + 3.06i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 59.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-3.18 + 3.18i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (5.60 + 20.9i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (19.3 - 72.1i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (77.5 - 44.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-78.4 - 45.3i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (22.2 - 83.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 23.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-62.9 - 16.8i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-21.9 + 38.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (105. + 105. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (34.4 - 59.6i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (60.9 + 60.9i)T + 9.40e3iT^{2} \) |
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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
−12.49739817731108623573769007195, −11.77799900526438192742318070817, −10.57160653822406613220132322229, −9.319443845886757537622711238518, −8.442745171276307640717830351514, −7.21485904738232512650365683403, −5.70422606139718304260111050270, −4.30968322386098487501201311558, −2.16431655922947833142009727161, −1.39293599219438885477933565466,
2.97590459753496471562474425212, 4.72884609911608537575462670598, 5.30275660255637751717754967803, 6.86716700665740399031228773418, 8.133872899970885379450949033720, 9.381853320233341464117717544901, 9.823090314528036457902797027730, 11.04664732088506091966787144068, 12.56089372508370152524103275940, 13.89898850495587914613318362052
