L(s) = 1 | + (2.89 − 1.67i)2-s + (2.59 + 1.5i)3-s + (1.58 − 2.73i)4-s + (0.879 + 11.1i)5-s + 10.0·6-s + (6.15 + 17.4i)7-s + 16.1i·8-s + (4.5 + 7.79i)9-s + (21.1 + 30.7i)10-s + (17.8 − 30.8i)11-s + (8.21 − 4.74i)12-s − 88.7i·13-s + (46.9 + 40.2i)14-s + (−14.4 + 30.2i)15-s + (39.6 + 68.6i)16-s + (−84.5 − 48.8i)17-s + ⋯ |
L(s) = 1 | + (1.02 − 0.590i)2-s + (0.499 + 0.288i)3-s + (0.197 − 0.342i)4-s + (0.0786 + 0.996i)5-s + 0.681·6-s + (0.332 + 0.943i)7-s + 0.714i·8-s + (0.166 + 0.288i)9-s + (0.669 + 0.973i)10-s + (0.488 − 0.845i)11-s + (0.197 − 0.114i)12-s − 1.89i·13-s + (0.897 + 0.768i)14-s + (−0.248 + 0.521i)15-s + (0.619 + 1.07i)16-s + (−1.20 − 0.696i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.346i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.98259 + 0.533903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.98259 + 0.533903i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.59 - 1.5i)T \) |
| 5 | \( 1 + (-0.879 - 11.1i)T \) |
| 7 | \( 1 + (-6.15 - 17.4i)T \) |
good | 2 | \( 1 + (-2.89 + 1.67i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-17.8 + 30.8i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 88.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (84.5 + 48.8i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-11.6 - 20.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-92.0 + 53.1i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 166.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (20.1 - 34.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (42.1 - 24.3i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (363. - 209. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-51.3 - 29.6i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (184. - 319. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (465. + 806. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (257. + 148. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 715.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (709. + 409. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (108. + 188. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 194. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-299. - 517. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.49e3iT - 9.12e5T^{2} \) |
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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
−13.42648167068808433801802832223, −12.39336131726902005403543080413, −11.26377284729246846226916180586, −10.59122469020253977693984654672, −8.980374701366953843640852168195, −7.946816926747869555667421095134, −6.14511525276557164111009061703, −4.94255318680373751318630046343, −3.28944663991435101811328267086, −2.59076783760239737956575515689,
1.49971922709741780874437850299, 4.17744488012442661895964265739, 4.62517353746957508901141769544, 6.46387723440785303031285032807, 7.28451979562296302415908728302, 8.831959838240491568636709875376, 9.734922628951627501466514314950, 11.48038304960784778539948505436, 12.65073418130863626390634908715, 13.43165874959645049013787696924