L(s) = 1 | + (0.550 − 0.147i)2-s + (−1.06 + 0.616i)3-s + (−1.45 + 0.837i)4-s + (−0.377 − 1.41i)5-s + (−0.496 + 0.496i)6-s + (−1.48 + 1.48i)8-s + (−0.740 + 1.28i)9-s + (−0.416 − 0.720i)10-s + (0.815 + 0.218i)11-s + (1.03 − 1.78i)12-s + (3.59 − 0.296i)13-s + (1.27 + 1.27i)15-s + (1.07 − 1.86i)16-s + (−3.67 − 6.36i)17-s + (−0.218 + 0.815i)18-s + (−1.31 − 4.90i)19-s + ⋯ |
L(s) = 1 | + (0.389 − 0.104i)2-s + (−0.616 + 0.355i)3-s + (−0.725 + 0.418i)4-s + (−0.168 − 0.630i)5-s + (−0.202 + 0.202i)6-s + (−0.523 + 0.523i)8-s + (−0.246 + 0.427i)9-s + (−0.131 − 0.227i)10-s + (0.245 + 0.0658i)11-s + (0.297 − 0.516i)12-s + (0.996 − 0.0822i)13-s + (0.328 + 0.328i)15-s + (0.269 − 0.466i)16-s + (−0.891 − 1.54i)17-s + (−0.0515 + 0.192i)18-s + (−0.301 − 1.12i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.413916 - 0.480345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.413916 - 0.480345i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.59 + 0.296i)T \) |
good | 2 | \( 1 + (-0.550 + 0.147i)T + (1.73 - i)T^{2} \) |
| 3 | \( 1 + (1.06 - 0.616i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.377 + 1.41i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.815 - 0.218i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (3.67 + 6.36i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.31 + 4.90i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (3.84 + 2.22i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 + (1.73 + 0.465i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.05 - 3.93i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.23 - 1.23i)T - 41iT^{2} \) |
| 43 | \( 1 + 8.66iT - 43T^{2} \) |
| 47 | \( 1 + (-3.44 + 0.923i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.89 - 8.48i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.398 + 1.48i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (6.15 + 3.55i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.20 + 11.9i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (1.46 + 1.46i)T + 71iT^{2} \) |
| 73 | \( 1 + (-0.0380 + 0.141i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.39 + 4.15i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (12.3 - 12.3i)T - 83iT^{2} \) |
| 89 | \( 1 + (9.41 - 2.52i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-6.05 + 6.05i)T - 97iT^{2} \) |
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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
−10.52791968401810037899147617194, −9.228340517238785459704529673594, −8.804540689526323550125476415999, −7.86572808268549169725397892801, −6.59084350410297969281752347100, −5.43021050716719083851694794915, −4.75349163241138899961746681830, −4.04470035028662503593259632780, −2.61823989636629772614419716788, −0.34977265039633772082168814214,
1.49236353395139777768413500079, 3.54416066091529137197658415680, 4.15082109037953640578493537061, 5.77062425661599354519953518671, 6.03539054709593630093056442702, 6.94992868495053652978305961680, 8.310992946842598824344196024695, 9.005401699626988610577182520103, 10.11232408055794931372123519222, 10.89743332338797739836351603272