| L(s) = 1 | + (1.39 + 0.221i)2-s + (−0.299 − 0.587i)3-s + (1.90 + 0.618i)4-s + (3.77 + 3.28i)5-s + (−0.288 − 0.886i)6-s + (−5.08 − 5.08i)7-s + (2.52 + 1.28i)8-s + (5.03 − 6.92i)9-s + (4.54 + 5.41i)10-s + (−15.3 + 11.1i)11-s + (−0.206 − 1.30i)12-s + (−15.1 + 2.39i)13-s + (−5.97 − 8.21i)14-s + (0.798 − 3.19i)15-s + (3.23 + 2.35i)16-s + (2.06 − 4.04i)17-s + ⋯ |
| L(s) = 1 | + (0.698 + 0.110i)2-s + (−0.0997 − 0.195i)3-s + (0.475 + 0.154i)4-s + (0.754 + 0.656i)5-s + (−0.0480 − 0.147i)6-s + (−0.725 − 0.725i)7-s + (0.315 + 0.160i)8-s + (0.559 − 0.769i)9-s + (0.454 + 0.541i)10-s + (−1.39 + 1.01i)11-s + (−0.0171 − 0.108i)12-s + (−1.16 + 0.184i)13-s + (−0.426 − 0.587i)14-s + (0.0532 − 0.213i)15-s + (0.202 + 0.146i)16-s + (0.121 − 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.58063 + 0.0848824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58063 + 0.0848824i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.221i)T \) |
| 5 | \( 1 + (-3.77 - 3.28i)T \) |
| good | 3 | \( 1 + (0.299 + 0.587i)T + (-5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (5.08 + 5.08i)T + 49iT^{2} \) |
| 11 | \( 1 + (15.3 - 11.1i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (15.1 - 2.39i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-2.06 + 4.04i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-7.61 + 2.47i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-4.63 + 29.2i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-41.1 - 13.3i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-7.78 - 23.9i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (1.63 + 10.3i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-31.9 - 23.2i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (16.0 - 16.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-14.0 + 7.16i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (22.4 + 43.9i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (8.28 - 11.4i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (52.4 - 38.0i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-1.12 + 2.19i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-34.1 + 105. i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (20.0 - 126. i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (52.5 + 17.0i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-132. - 67.7i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (58.7 + 80.8i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-74.7 + 38.0i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−15.15254051656735287105832642258, −14.17521885684518072481003697087, −13.02102607073092372216964053669, −12.32094382651323006481156957462, −10.45133677626366857992718469607, −9.800059948686003779165442958555, −7.32072333607455648429687590488, −6.59610396491377569047057408371, −4.82602425428468566125798403411, −2.77970588471608344981312068681,
2.64635720891888758263036438852, 4.99683286086406744735960882616, 5.86871482382890854389150549831, 7.84642649986921661692881451799, 9.566986394201030538051932084536, 10.54100635391084567062677232699, 12.16842808508516136328366600685, 13.12007192768149450925601413450, 13.81389511049371749975178976032, 15.52491610690331396175249829683
