| L(s) = 1 | + (−0.368 − 1.37i)2-s + (1.70 − 0.983i)4-s + (−2.49 + 9.31i)5-s + (−4.83 − 8.37i)7-s + (−6.01 − 6.01i)8-s + 13.7·10-s + 0.381i·11-s + (−4.47 + 16.6i)13-s + (−9.75 + 9.75i)14-s + (−2.12 + 3.68i)16-s + (−10.5 + 2.81i)17-s + (0.381 − 1.42i)19-s + (4.91 + 18.3i)20-s + (0.524 − 0.140i)22-s + (−22.0 − 22.0i)23-s + ⋯ |
| L(s) = 1 | + (−0.184 − 0.688i)2-s + (0.426 − 0.245i)4-s + (−0.499 + 1.86i)5-s + (−0.691 − 1.19i)7-s + (−0.751 − 0.751i)8-s + 1.37·10-s + 0.0346i·11-s + (−0.343 + 1.28i)13-s + (−0.696 + 0.696i)14-s + (−0.133 + 0.230i)16-s + (−0.618 + 0.165i)17-s + (0.0200 − 0.0748i)19-s + (0.245 + 0.916i)20-s + (0.0238 − 0.00639i)22-s + (−0.960 − 0.960i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.105098 + 0.214600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105098 + 0.214600i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 37 | \( 1 + (-34.6 - 12.8i)T \) |
| good | 2 | \( 1 + (0.368 + 1.37i)T + (-3.46 + 2i)T^{2} \) |
| 5 | \( 1 + (2.49 - 9.31i)T + (-21.6 - 12.5i)T^{2} \) |
| 7 | \( 1 + (4.83 + 8.37i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 0.381iT - 121T^{2} \) |
| 13 | \( 1 + (4.47 - 16.6i)T + (-146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (10.5 - 2.81i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-0.381 + 1.42i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (22.0 + 22.0i)T + 529iT^{2} \) |
| 29 | \( 1 + (25.2 - 25.2i)T - 841iT^{2} \) |
| 31 | \( 1 + (35.4 - 35.4i)T - 961iT^{2} \) |
| 41 | \( 1 + (36.4 - 21.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-20.0 - 20.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 44.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-14.7 + 25.4i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-23.5 + 6.31i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (16.2 + 4.35i)T + (3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (-17.8 + 10.3i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (28.3 + 49.1i)T + (-2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + 56.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-5.87 + 21.9i)T + (-5.40e3 - 3.12e3i)T^{2} \) |
| 83 | \( 1 + (-54.9 + 95.1i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-5.79 - 21.6i)T + (-6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-38.1 - 38.1i)T + 9.40e3iT^{2} \) |
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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
−11.37598377203907767910425079049, −10.69690841633702686034485550221, −10.26308095509844219972371993311, −9.290450443140405868055978976171, −7.51548622420424184560636943068, −6.79606159092268217912359035682, −6.38636452754668255878498770591, −4.07760057096572924455046122261, −3.26849370660620221567111723950, −2.09125688648665571086974194734,
0.10622792281224019322097897838, 2.29487877036559486385193933086, 3.89658933472394802296824229632, 5.59893913465103686067673789691, 5.68865879213459089078007740621, 7.47211730737439314660634356671, 8.149291990348302558962342628963, 8.975364924277678940947242044329, 9.615035974050711801785016578012, 11.36238153285853587382382560735
