L(s) = 1 | + (−2.18 + 1.25i)5-s + (3.00 + 6.31i)7-s + (2.78 − 4.81i)11-s + 6.50i·13-s + (17.6 + 10.1i)17-s + (−9.98 + 5.76i)19-s + (−16.3 − 28.2i)23-s + (−9.32 + 16.1i)25-s + 19.7·29-s + (5.23 + 3.02i)31-s + (−14.5 − 9.99i)35-s + (−5.71 − 9.89i)37-s + 69.9i·41-s − 40.6·43-s + (33.5 − 19.3i)47-s + ⋯ |
L(s) = 1 | + (−0.436 + 0.251i)5-s + (0.429 + 0.902i)7-s + (0.252 − 0.437i)11-s + 0.500i·13-s + (1.03 + 0.597i)17-s + (−0.525 + 0.303i)19-s + (−0.709 − 1.22i)23-s + (−0.373 + 0.646i)25-s + 0.679·29-s + (0.168 + 0.0974i)31-s + (−0.415 − 0.285i)35-s + (−0.154 − 0.267i)37-s + 1.70i·41-s − 0.945·43-s + (0.713 − 0.411i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.263308053\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.263308053\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-3.00 - 6.31i)T \) |
good | 5 | \( 1 + (2.18 - 1.25i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.78 + 4.81i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 6.50iT - 169T^{2} \) |
| 17 | \( 1 + (-17.6 - 10.1i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (9.98 - 5.76i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (16.3 + 28.2i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 19.7T + 841T^{2} \) |
| 31 | \( 1 + (-5.23 - 3.02i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (5.71 + 9.89i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 69.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-33.5 + 19.3i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (16.4 - 28.5i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (61.4 + 35.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (35.8 - 20.7i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (47.9 - 82.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 57.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-28.2 - 16.3i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-31.1 - 53.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 35.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-6.60 + 3.81i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 130. iT - 9.40e3T^{2} \) |
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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
−10.11042472413291088790601232883, −9.104399305695014761248140512155, −8.331694787227227203230584199260, −7.77089266752801509165043034532, −6.51123328949789715520084909443, −5.89526371583201835575031646693, −4.78923849400104517245338930335, −3.80168722584035941748963250542, −2.71319153703638514845193479347, −1.46827874232569200613348128202,
0.40695423232611017038036296031, 1.68146899684116583924509556192, 3.23392684216337832332601139150, 4.17996140115946368928214960846, 4.97850125663866701226983494911, 6.04683415748670662398859077463, 7.20988069042870267699236813120, 7.73324946768779795054290235497, 8.517434306196631604620488308980, 9.627672733020159020451924486793