| L(s) = 1 | + (−0.830 − 1.81i)2-s + (−4.95 + 1.45i)3-s + (−2.61 + 3.02i)4-s + (5.53 + 3.55i)5-s + (6.76 + 7.80i)6-s + (4.17 + 29.0i)7-s + (7.67 + 2.25i)8-s + (−0.274 + 0.176i)9-s + (1.87 − 13.0i)10-s + (−6.42 + 14.0i)11-s + (8.58 − 18.7i)12-s + (0.215 − 1.49i)13-s + (49.3 − 31.7i)14-s + (−32.6 − 9.57i)15-s + (−2.27 − 15.8i)16-s + (−32.0 − 37.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.953 + 0.280i)3-s + (−0.327 + 0.377i)4-s + (0.495 + 0.318i)5-s + (0.460 + 0.531i)6-s + (0.225 + 1.56i)7-s + (0.339 + 0.0996i)8-s + (−0.0101 + 0.00652i)9-s + (0.0592 − 0.411i)10-s + (−0.176 + 0.385i)11-s + (0.206 − 0.452i)12-s + (0.00460 − 0.0319i)13-s + (0.941 − 0.605i)14-s + (−0.561 − 0.164i)15-s + (−0.0355 − 0.247i)16-s + (−0.457 − 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.584310 + 0.428900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.584310 + 0.428900i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.830 + 1.81i)T \) |
| 23 | \( 1 + (-100. - 45.1i)T \) |
| good | 3 | \( 1 + (4.95 - 1.45i)T + (22.7 - 14.5i)T^{2} \) |
| 5 | \( 1 + (-5.53 - 3.55i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (-4.17 - 29.0i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (6.42 - 14.0i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-0.215 + 1.49i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (32.0 + 37.0i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (76.4 - 88.2i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (-82.5 - 95.2i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (-64.6 - 18.9i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-172. + 110. i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (-281. - 180. i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (452. - 132. i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 - 388.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (43.2 + 300. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-49.9 + 347. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-63.5 - 18.6i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (329. + 721. i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-410. - 898. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (-396. + 457. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (171. - 1.19e3i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-186. + 119. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (962. - 282. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-1.28e3 - 828. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−15.63939473552478244627493312256, −14.39884752024105018687957186610, −12.75734134015650973435737677412, −11.82178520671217993211753111374, −10.88164250647962932800643726342, −9.699885104135618500044980138885, −8.405456809026336974833321507783, −6.21049818104004173258759211894, −4.99187828897979652401321791942, −2.41156552487014398587817636398,
0.71345301801139557944856674321, 4.60846183573063617182838627214, 6.12006060579123165192759398180, 7.18430236419770820344721809397, 8.769012616563514000612849650521, 10.38951111551795048565968828914, 11.23230040756396270192855822490, 12.99694094174744129275935999033, 13.78969548725740894914933757962, 15.18841198049161382534471776058
