| L(s) = 1 | + (−0.100 − 1.41i)2-s + (−2.42 − 0.526i)3-s + (−1.97 + 0.284i)4-s + (−4.73 − 1.61i)5-s + (−0.498 + 3.46i)6-s + (0.940 − 1.72i)7-s + (0.601 + 2.76i)8-s + (−2.60 − 1.18i)9-s + (−1.80 + 6.83i)10-s + (2.86 + 3.30i)11-s + (4.94 + 0.353i)12-s + (6.26 + 11.4i)13-s + (−2.52 − 1.15i)14-s + (10.6 + 6.41i)15-s + (3.83 − 1.12i)16-s + (7.48 + 9.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 − 0.705i)2-s + (−0.807 − 0.175i)3-s + (−0.494 + 0.0711i)4-s + (−0.946 − 0.323i)5-s + (−0.0831 + 0.578i)6-s + (0.134 − 0.245i)7-s + (0.0751 + 0.345i)8-s + (−0.289 − 0.132i)9-s + (−0.180 + 0.683i)10-s + (0.260 + 0.300i)11-s + (0.411 + 0.0294i)12-s + (0.481 + 0.882i)13-s + (−0.180 − 0.0823i)14-s + (0.706 + 0.427i)15-s + (0.239 − 0.0704i)16-s + (0.440 + 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.438385 + 0.190760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.438385 + 0.190760i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.100 + 1.41i)T \) |
| 5 | \( 1 + (4.73 + 1.61i)T \) |
| 23 | \( 1 + (10.9 - 20.2i)T \) |
| good | 3 | \( 1 + (2.42 + 0.526i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (-0.940 + 1.72i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (-2.86 - 3.30i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-6.26 - 11.4i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (-7.48 - 9.99i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (-0.916 + 0.131i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (13.0 + 1.88i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-3.80 - 2.44i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (-9.48 - 25.4i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (25.0 + 54.8i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (32.3 + 7.03i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (36.2 - 36.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-0.322 - 0.176i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (20.7 - 70.6i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (52.0 + 33.4i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-7.29 - 102. i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (9.82 - 11.3i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-22.1 + 29.6i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (-8.10 + 27.6i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (136. - 50.8i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (44.7 + 69.6i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-71.4 - 26.6i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−11.84458825679966371951819584168, −11.44109098892931147842748245865, −10.48734584664432258201747868750, −9.230487979293375129066049960869, −8.275094168969591745067171689121, −7.10211558258266961341049023120, −5.80825161195679099292225949495, −4.52979676158214343050679624516, −3.50737030022315682914542715976, −1.34276382872477427241257457241,
0.32518728332623754071265080683, 3.26258631929892586135943085983, 4.69099857817565457647278189023, 5.71136642841170575805764943324, 6.67105781537817321982932929540, 7.908936554140667703205592479290, 8.563175887878589749475082546995, 10.03072031600404228648083547610, 11.02835498868129315009348003018, 11.71684605650376680630559180028
