| L(s) = 1 | + (1.06 − 1.32i)2-s + (−1.14 + 0.172i)3-s + (−0.198 − 0.868i)4-s + (−1.45 + 0.993i)5-s + (−0.982 + 1.70i)6-s + (−0.297 − 0.514i)7-s + (1.69 + 0.818i)8-s + (−1.59 + 0.490i)9-s + (−0.224 + 2.99i)10-s + (0.967 − 4.23i)11-s + (0.376 + 0.958i)12-s + (0.242 + 3.23i)13-s + (−0.999 − 0.150i)14-s + (1.49 − 1.38i)15-s + (4.49 − 2.16i)16-s + (−4.57 − 3.12i)17-s + ⋯ |
| L(s) = 1 | + (0.749 − 0.940i)2-s + (−0.659 + 0.0994i)3-s + (−0.0991 − 0.434i)4-s + (−0.651 + 0.444i)5-s + (−0.401 + 0.694i)6-s + (−0.112 − 0.194i)7-s + (0.600 + 0.289i)8-s + (−0.530 + 0.163i)9-s + (−0.0708 + 0.945i)10-s + (0.291 − 1.27i)11-s + (0.108 + 0.276i)12-s + (0.0672 + 0.896i)13-s + (−0.267 − 0.0402i)14-s + (0.385 − 0.357i)15-s + (1.12 − 0.541i)16-s + (−1.11 − 0.757i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.810762 - 0.339934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.810762 - 0.339934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.98 - 4.26i)T \) |
| good | 2 | \( 1 + (-1.06 + 1.32i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (1.14 - 0.172i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (1.45 - 0.993i)T + (1.82 - 4.65i)T^{2} \) |
| 7 | \( 1 + (0.297 + 0.514i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.967 + 4.23i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.242 - 3.23i)T + (-12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (4.57 + 3.12i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-2.59 - 0.801i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (3.88 + 3.60i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-9.75 - 1.46i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (-1.36 - 3.47i)T + (-22.7 + 21.0i)T^{2} \) |
| 37 | \( 1 + (0.673 - 1.16i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.99 - 8.76i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (0.909 + 3.98i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (0.180 - 2.40i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-0.662 + 0.318i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 4.76i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (3.67 + 1.13i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (-2.81 + 2.61i)T + (5.30 - 70.8i)T^{2} \) |
| 73 | \( 1 + (0.557 + 7.43i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (6.69 + 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.70 - 1.16i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (4.45 - 0.672i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (0.0411 - 0.180i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−16.04189666182554208850152130134, −14.23771126239141191582669169873, −13.55722148014165654791634395218, −11.77240214215309035280517344032, −11.56720139572277705383436437833, −10.47440846171300761437015464409, −8.413380881369651171420779469201, −6.55755174385236739612120406500, −4.73122636797351785758488045135, −3.17168709967402722657061863269,
4.34580876017382499711072222196, 5.64535937000768248828332868079, 6.88150801314152378769871128841, 8.303490895530590148022783873285, 10.22309001576513647948732318770, 11.80685293274997041639591284242, 12.68348610944047794647015429328, 14.00097670270208600860755329404, 15.37367986182524400340334957855, 15.72092194761325653714425755269
