| L(s) = 1 | + (1.35 + 1.86i)2-s + (−0.117 + 2.20i)3-s + (−1.00 + 3.14i)4-s + (−0.577 + 0.0123i)5-s + (−4.25 + 2.77i)6-s + (2.61 + 0.391i)7-s + (−2.85 + 0.948i)8-s + (−1.85 − 0.199i)9-s + (−0.807 − 1.05i)10-s + (−0.0665 + 0.563i)11-s + (−6.81 − 2.59i)12-s + (−2.43 − 4.67i)13-s + (2.82 + 5.40i)14-s + (0.0408 − 1.27i)15-s + (−0.260 − 0.186i)16-s + (−0.707 + 0.169i)17-s + ⋯ |
| L(s) = 1 | + (0.960 + 1.31i)2-s + (−0.0680 + 1.27i)3-s + (−0.504 + 1.57i)4-s + (−0.258 + 0.00552i)5-s + (−1.73 + 1.13i)6-s + (0.989 + 0.147i)7-s + (−1.00 + 0.335i)8-s + (−0.618 − 0.0663i)9-s + (−0.255 − 0.334i)10-s + (−0.0200 + 0.170i)11-s + (−1.96 − 0.748i)12-s + (−0.676 − 1.29i)13-s + (0.755 + 1.44i)14-s + (0.0105 − 0.329i)15-s + (−0.0652 − 0.0465i)16-s + (−0.171 + 0.0410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.184319 + 2.13947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.184319 + 2.13947i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.61 - 0.391i)T \) |
| good | 2 | \( 1 + (-1.35 - 1.86i)T + (-0.609 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.117 - 2.20i)T + (-2.98 - 0.319i)T^{2} \) |
| 5 | \( 1 + (0.577 - 0.0123i)T + (4.99 - 0.213i)T^{2} \) |
| 11 | \( 1 + (0.0665 - 0.563i)T + (-10.6 - 2.56i)T^{2} \) |
| 13 | \( 1 + (2.43 + 4.67i)T + (-7.43 + 10.6i)T^{2} \) |
| 17 | \( 1 + (0.707 - 0.169i)T + (15.1 - 7.70i)T^{2} \) |
| 19 | \( 1 + (-3.91 + 6.78i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.394 + 0.307i)T + (5.59 - 22.3i)T^{2} \) |
| 29 | \( 1 + (-2.47 + 1.00i)T + (20.8 - 20.1i)T^{2} \) |
| 31 | \( 1 + (1.15 - 1.06i)T + (2.31 - 30.9i)T^{2} \) |
| 37 | \( 1 + (0.610 + 0.644i)T + (-1.97 + 36.9i)T^{2} \) |
| 41 | \( 1 + (1.20 - 3.28i)T + (-31.2 - 26.5i)T^{2} \) |
| 43 | \( 1 + (-7.22 - 2.39i)T + (34.4 + 25.7i)T^{2} \) |
| 47 | \( 1 + (9.79 + 5.24i)T + (26.0 + 39.1i)T^{2} \) |
| 53 | \( 1 + (-5.94 - 3.51i)T + (25.5 + 46.4i)T^{2} \) |
| 59 | \( 1 + (2.38 - 2.86i)T + (-10.6 - 58.0i)T^{2} \) |
| 61 | \( 1 + (11.4 - 0.981i)T + (60.1 - 10.3i)T^{2} \) |
| 67 | \( 1 + (-1.96 + 0.604i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (7.04 + 2.85i)T + (51.0 + 49.3i)T^{2} \) |
| 73 | \( 1 + (0.862 - 0.461i)T + (40.4 - 60.7i)T^{2} \) |
| 79 | \( 1 + (2.84 - 0.428i)T + (75.4 - 23.2i)T^{2} \) |
| 83 | \( 1 + (-14.7 + 2.87i)T + (76.9 - 31.1i)T^{2} \) |
| 89 | \( 1 + (-2.62 + 2.65i)T + (-0.951 - 88.9i)T^{2} \) |
| 97 | \( 1 + (3.62 + 15.8i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−12.03640210256269638454458981671, −11.04856939599311885528915769503, −10.11107475072830563441942328630, −8.988135600969142326481163261252, −7.904742088490239401203897618563, −7.22614981828330216779602106987, −5.73572617736284799110306647715, −4.94388951494114148739811345517, −4.44966224581287571995386741301, −3.14046260527164199645117445760,
1.39487331164047660119175630274, 2.19511038800245920108194621721, 3.79000211047122483165767392633, 4.81628774757590715605655576709, 5.98243227351956454764676645678, 7.30915654054036715138421501666, 8.031038371391939713760592840393, 9.512672032983105741271514610179, 10.58934852718758017647952248165, 11.71266406027657747042114644231
