| L(s) = 1 | + (−0.900 + 0.433i)2-s + (0.183 + 2.45i)3-s + (0.623 − 0.781i)4-s + (−0.955 + 0.294i)5-s + (−1.22 − 2.12i)6-s + (−1.84 + 3.19i)7-s + (−0.222 + 0.974i)8-s + (−3.00 + 0.452i)9-s + (0.733 − 0.680i)10-s + (−0.117 − 0.147i)11-s + (2.03 + 1.38i)12-s + (−4.88 − 4.53i)13-s + (0.275 − 3.67i)14-s + (−0.897 − 2.28i)15-s + (−0.222 − 0.974i)16-s + (2.38 + 0.735i)17-s + ⋯ |
| L(s) = 1 | + (−0.637 + 0.306i)2-s + (0.106 + 1.41i)3-s + (0.311 − 0.390i)4-s + (−0.427 + 0.131i)5-s + (−0.501 − 0.868i)6-s + (−0.696 + 1.20i)7-s + (−0.0786 + 0.344i)8-s + (−1.00 + 0.150i)9-s + (0.231 − 0.215i)10-s + (−0.0355 − 0.0445i)11-s + (0.586 + 0.399i)12-s + (−1.35 − 1.25i)13-s + (0.0736 − 0.982i)14-s + (−0.231 − 0.590i)15-s + (−0.0556 − 0.243i)16-s + (0.578 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.152693 - 0.465469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.152693 - 0.465469i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.900 - 0.433i)T \) |
| 5 | \( 1 + (0.955 - 0.294i)T \) |
| 43 | \( 1 + (6.40 + 1.40i)T \) |
| good | 3 | \( 1 + (-0.183 - 2.45i)T + (-2.96 + 0.447i)T^{2} \) |
| 7 | \( 1 + (1.84 - 3.19i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.117 + 0.147i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (4.88 + 4.53i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-2.38 - 0.735i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-3.06 - 0.462i)T + (18.1 + 5.60i)T^{2} \) |
| 23 | \( 1 + (3.04 - 7.77i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.432 + 5.76i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (5.19 + 3.54i)T + (11.3 + 28.8i)T^{2} \) |
| 37 | \( 1 + (-1.60 - 2.78i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.60 - 2.69i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-8.12 + 10.1i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (1.08 - 1.01i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-3.02 - 13.2i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (2.76 - 1.88i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (8.04 + 1.21i)T + (64.0 + 19.7i)T^{2} \) |
| 71 | \( 1 + (-4.13 - 10.5i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (2.93 + 2.72i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (3.74 - 6.48i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.543 - 7.25i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-1.20 - 16.0i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-8.12 - 10.1i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.63310936108524629622812535901, −10.32472597417005133838745934832, −9.841894367787665212204550636969, −9.266771548758690012249956973885, −8.189626926102852355726363933026, −7.33626901687440019367662122401, −5.72016820606253606586339251679, −5.26752642421692352932397454802, −3.68663644479762053550252350772, −2.69998793806974254422319137429,
0.36393081880216831604050728954, 1.79607956578022501499570701511, 3.19231408668577586350083303588, 4.59022770632399339098964453020, 6.45804944824384696888281465396, 7.22287363699630470939107083711, 7.51796721046275387096910465366, 8.717645652952454276037051291385, 9.728693995625747810738666853992, 10.56843150893510171147572631602
