| L(s) = 1 | + (0.930 + 0.365i)2-s + (1.34 − 1.09i)3-s + (0.733 + 0.680i)4-s + (−3.54 − 2.41i)5-s + (1.65 − 0.524i)6-s + (0.973 − 2.46i)7-s + (0.433 + 0.900i)8-s + (0.618 − 2.93i)9-s + (−2.41 − 3.54i)10-s + (0.317 + 2.10i)11-s + (1.72 + 0.114i)12-s + (0.621 − 0.495i)13-s + (1.80 − 1.93i)14-s + (−7.41 + 0.618i)15-s + (0.0747 + 0.997i)16-s + (7.16 + 2.21i)17-s + ⋯ |
| L(s) = 1 | + (0.658 + 0.258i)2-s + (0.776 − 0.630i)3-s + (0.366 + 0.340i)4-s + (−1.58 − 1.08i)5-s + (0.673 − 0.214i)6-s + (0.367 − 0.929i)7-s + (0.153 + 0.318i)8-s + (0.206 − 0.978i)9-s + (−0.765 − 1.12i)10-s + (0.0957 + 0.635i)11-s + (0.498 + 0.0331i)12-s + (0.172 − 0.137i)13-s + (0.482 − 0.516i)14-s + (−1.91 + 0.159i)15-s + (0.0186 + 0.249i)16-s + (1.73 + 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70704 - 0.922672i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70704 - 0.922672i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.930 - 0.365i)T \) |
| 3 | \( 1 + (-1.34 + 1.09i)T \) |
| 7 | \( 1 + (-0.973 + 2.46i)T \) |
| good | 5 | \( 1 + (3.54 + 2.41i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.317 - 2.10i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-0.621 + 0.495i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-7.16 - 2.21i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (1.88 - 1.08i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.36 - 4.43i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (6.59 - 1.50i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.56 + 1.48i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.202 + 0.187i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-5.29 + 2.54i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.05 - 1.95i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (0.177 - 0.451i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-3.48 + 3.75i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-3.31 + 2.26i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (1.09 + 1.18i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.16 + 3.75i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.36 - 0.540i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (11.5 - 4.52i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-5.04 - 8.74i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.49 - 9.39i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (5.32 + 0.803i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 3.52iT - 97T^{2} \) |
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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
−11.99927249761368912518313432871, −11.04937872703631124363019498454, −9.517533479811425931192947541025, −8.300705014421327723272632424245, −7.68828393082573082865471406920, −7.18918362379481073128004045646, −5.42412472650619859643802655989, −4.07123723101514385114716364575, −3.60710530435804225854873142675, −1.30048789037884062168529084630,
2.68060236393400193227796475766, 3.43815802724841039355176605681, 4.41960179528052652508697074232, 5.75108046102807883747322442404, 7.25631127676510160922030518829, 8.032762455889741229730903510258, 9.003788858882179921858684371134, 10.34925610715215602424444066255, 11.13707415528664252483874027891, 11.76046521481850453274485202722
