| L(s) = 1 | + (−0.122 + 1.99i)2-s + (−1.35 − 1.07i)3-s + (−3.96 − 0.489i)4-s + (4.94 + 2.38i)5-s + (2.32 − 2.57i)6-s + (−3.10 − 2.47i)7-s + (1.46 − 7.86i)8-s + (0.667 + 2.92i)9-s + (−5.35 + 9.57i)10-s + (−9.56 − 2.18i)11-s + (4.84 + 4.95i)12-s + (−1.79 + 7.88i)13-s + (5.32 − 5.90i)14-s + (−4.12 − 8.56i)15-s + (15.5 + 3.88i)16-s − 19.7·17-s + ⋯ |
| L(s) = 1 | + (−0.0613 + 0.998i)2-s + (−0.451 − 0.359i)3-s + (−0.992 − 0.122i)4-s + (0.988 + 0.476i)5-s + (0.386 − 0.428i)6-s + (−0.444 − 0.354i)7-s + (0.183 − 0.983i)8-s + (0.0741 + 0.324i)9-s + (−0.535 + 0.957i)10-s + (−0.869 − 0.198i)11-s + (0.403 + 0.412i)12-s + (−0.138 + 0.606i)13-s + (0.380 − 0.421i)14-s + (−0.274 − 0.570i)15-s + (0.970 + 0.243i)16-s − 1.16·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0174075 - 0.0293123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0174075 - 0.0293123i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.122 - 1.99i)T \) |
| 3 | \( 1 + (1.35 + 1.07i)T \) |
| 29 | \( 1 + (-7.10 + 28.1i)T \) |
| good | 5 | \( 1 + (-4.94 - 2.38i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (3.10 + 2.47i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (9.56 + 2.18i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (1.79 - 7.88i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + 19.7T + 289T^{2} \) |
| 19 | \( 1 + (16.9 - 13.5i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (6.60 + 13.7i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (0.210 - 0.437i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (8.38 + 36.7i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 + 59.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (10.9 + 22.7i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (31.3 + 7.15i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (-27.8 - 13.4i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 - 82.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (44.7 - 56.1i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (-82.1 + 18.7i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (15.2 + 3.46i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (59.7 - 28.7i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (-127. + 29.0i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (46.9 - 37.4i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-50.3 - 24.2i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (-85.6 - 107. i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−10.55863231092834720603871557722, −10.16057084456391667267275463757, −9.003298724981757443091694658761, −7.990975519490647657037339035930, −6.78406088690335964415569554969, −6.35452088696307946735396832878, −5.37491341246105978689590176025, −4.12705748150074565992178393072, −2.20083057836710955810880960933, −0.01580907249692519749186693790,
1.88113063483855683019430127822, 3.10636443420521897129521237160, 4.72097388334919935816890366969, 5.34889227700177899768942165319, 6.52754753049941992384772077610, 8.248139102212712280524590997707, 9.149954490661319682744906601065, 9.879904071699725833236577147805, 10.59380687640332852746233058134, 11.43932376245362168073615033014
