| L(s) = 1 | + (0.930 + 0.365i)2-s + (−0.332 + 1.69i)3-s + (0.733 + 0.680i)4-s + (2.77 + 1.89i)5-s + (−0.930 + 1.46i)6-s + (−2.14 + 1.54i)7-s + (0.433 + 0.900i)8-s + (−2.77 − 1.13i)9-s + (1.89 + 2.77i)10-s + (−0.905 − 6.00i)11-s + (−1.40 + 1.01i)12-s + (1.32 − 1.05i)13-s + (−2.56 + 0.654i)14-s + (−4.13 + 4.08i)15-s + (0.0747 + 0.997i)16-s + (0.409 + 0.126i)17-s + ⋯ |
| L(s) = 1 | + (0.658 + 0.258i)2-s + (−0.192 + 0.981i)3-s + (0.366 + 0.340i)4-s + (1.24 + 0.845i)5-s + (−0.380 + 0.596i)6-s + (−0.811 + 0.584i)7-s + (0.153 + 0.318i)8-s + (−0.926 − 0.377i)9-s + (0.597 + 0.876i)10-s + (−0.272 − 1.81i)11-s + (−0.404 + 0.294i)12-s + (0.368 − 0.293i)13-s + (−0.685 + 0.174i)14-s + (−1.06 + 1.05i)15-s + (0.0186 + 0.249i)16-s + (0.0993 + 0.0306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24891 + 1.43109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24891 + 1.43109i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.332 - 1.69i)T \) |
| 7 | \( 1 + (2.14 - 1.54i)T \) |
| good | 5 | \( 1 + (-2.77 - 1.89i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.905 + 6.00i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.32 + 1.05i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.409 - 0.126i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.985 + 0.568i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.51 - 4.91i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-7.31 + 1.66i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (1.31 + 0.757i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.65 - 7.09i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-9.57 + 4.61i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (6.51 + 3.13i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (1.97 - 5.02i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-0.516 + 0.556i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-7.56 + 5.15i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (3.47 + 3.74i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.86 + 6.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (13.2 + 3.03i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (1.42 - 0.559i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (2.37 + 4.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.83 - 3.54i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (3.81 + 0.574i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 4.18iT - 97T^{2} \) |
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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
−11.91439295857192743815842142056, −10.97823545659969838562459730624, −10.28231000081397317689493543271, −9.346086917395170069037974154458, −8.389836220924626993326374264572, −6.58056728979954717291288272693, −5.90834784907915250163803641240, −5.31985679954739630361702994059, −3.42232220396321996457580043112, −2.85513908887594953895694794263,
1.39526489147822424483950284294, 2.56995045283347349992887609149, 4.48806962449711463351906316971, 5.50663563497201220543405839464, 6.54308594545468339038923433994, 7.23497854650988105532331166593, 8.761822169126983624216772052003, 9.832891726823990022450447778830, 10.51395534303031938679746022413, 11.98299472508384751256655067491
