| L(s) = 1 | + (−0.582 + 2.55i)2-s + (−4.15 + 3.85i)3-s + (1.02 + 0.492i)4-s + (−13.6 − 2.05i)5-s + (−7.42 − 12.8i)6-s + (9.34 − 16.1i)7-s + (−14.9 + 18.7i)8-s + (0.380 − 5.07i)9-s + (13.2 − 33.6i)10-s + (−41.3 + 19.9i)11-s + (−6.14 + 1.89i)12-s + (13.3 + 33.9i)13-s + (35.9 + 33.3i)14-s + (64.5 − 44.0i)15-s + (−33.4 − 41.9i)16-s + (62.1 − 9.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.206 + 0.903i)2-s + (−0.799 + 0.741i)3-s + (0.127 + 0.0616i)4-s + (−1.22 − 0.183i)5-s + (−0.504 − 0.874i)6-s + (0.504 − 0.874i)7-s + (−0.659 + 0.827i)8-s + (0.0140 − 0.188i)9-s + (0.417 − 1.06i)10-s + (−1.13 + 0.545i)11-s + (−0.147 + 0.0456i)12-s + (0.284 + 0.724i)13-s + (0.685 + 0.636i)14-s + (1.11 − 0.758i)15-s + (−0.522 − 0.655i)16-s + (0.886 − 0.133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0538159 - 0.609622i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0538159 - 0.609622i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-16.1 - 281. i)T \) |
| good | 2 | \( 1 + (0.582 - 2.55i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (4.15 - 3.85i)T + (2.01 - 26.9i)T^{2} \) |
| 5 | \( 1 + (13.6 + 2.05i)T + (119. + 36.8i)T^{2} \) |
| 7 | \( 1 + (-9.34 + 16.1i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (41.3 - 19.9i)T + (829. - 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-13.3 - 33.9i)T + (-1.61e3 + 1.49e3i)T^{2} \) |
| 17 | \( 1 + (-62.1 + 9.36i)T + (4.69e3 - 1.44e3i)T^{2} \) |
| 19 | \( 1 + (-11.9 - 159. i)T + (-6.78e3 + 1.02e3i)T^{2} \) |
| 23 | \( 1 + (-30.1 - 20.5i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (0.0346 + 0.0321i)T + (1.82e3 + 2.43e4i)T^{2} \) |
| 31 | \( 1 + (-142. + 43.9i)T + (2.46e4 - 1.67e4i)T^{2} \) |
| 37 | \( 1 + (157. + 273. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (3.55 - 15.5i)T + (-6.20e4 - 2.99e4i)T^{2} \) |
| 47 | \( 1 + (-359. - 173. i)T + (6.47e4 + 8.11e4i)T^{2} \) |
| 53 | \( 1 + (-140. + 356. i)T + (-1.09e5 - 1.01e5i)T^{2} \) |
| 59 | \( 1 + (-326. - 409. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-315. - 97.3i)T + (1.87e5 + 1.27e5i)T^{2} \) |
| 67 | \( 1 + (13.4 + 179. i)T + (-2.97e5 + 4.48e4i)T^{2} \) |
| 71 | \( 1 + (343. - 233. i)T + (1.30e5 - 3.33e5i)T^{2} \) |
| 73 | \( 1 + (312. + 795. i)T + (-2.85e5 + 2.64e5i)T^{2} \) |
| 79 | \( 1 + (527. - 914. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-43.9 + 40.7i)T + (4.27e4 - 5.70e5i)T^{2} \) |
| 89 | \( 1 + (1.00e3 - 933. i)T + (5.26e4 - 7.02e5i)T^{2} \) |
| 97 | \( 1 + (712. - 343. i)T + (5.69e5 - 7.13e5i)T^{2} \) |
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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
−16.25697784595915796711941316408, −15.39633375031659341405691704298, −14.22379642324075952200623041055, −12.20845435985990374644543093012, −11.28594775080635695910461402223, −10.21707562788721932223581964742, −8.079611537465945198289068276696, −7.43253506126836834701001592859, −5.54795061870650394452019448718, −4.14888235372942873660968726123,
0.57429896065672115685097170996, 2.96322544168378304067210413393, 5.51320394113318502373891608767, 7.11751206037655239847800305666, 8.530613031274806532002090708591, 10.54819819964915286450610685474, 11.50273500556218469073876058200, 12.02272771383544111320623451308, 13.09446251155725769710153816671, 15.29540786012115499342995030464
