L(s) = 1 | + (0.422 − 0.906i)2-s + (−2.64 + 1.85i)3-s + (−0.642 − 0.766i)4-s + (0.560 + 3.17i)6-s + (−0.958 + 3.57i)7-s + (−0.965 + 0.258i)8-s + (2.53 − 6.96i)9-s + (−1.03 + 1.78i)11-s + (3.11 + 0.835i)12-s + (−1.25 + 1.78i)13-s + (2.83 + 2.37i)14-s + (−0.173 + 0.984i)16-s + (1.16 + 0.543i)17-s + (−5.24 − 5.24i)18-s + (−1.10 + 4.21i)19-s + ⋯ |
L(s) = 1 | + (0.298 − 0.640i)2-s + (−1.52 + 1.06i)3-s + (−0.321 − 0.383i)4-s + (0.228 + 1.29i)6-s + (−0.362 + 1.35i)7-s + (−0.341 + 0.0915i)8-s + (0.844 − 2.32i)9-s + (−0.311 + 0.538i)11-s + (0.899 + 0.241i)12-s + (−0.347 + 0.496i)13-s + (0.757 + 0.635i)14-s + (−0.0434 + 0.246i)16-s + (0.282 + 0.131i)17-s + (−1.23 − 1.23i)18-s + (−0.253 + 0.967i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00592117 - 0.0189414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00592117 - 0.0189414i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.422 + 0.906i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (1.10 - 4.21i)T \) |
good | 3 | \( 1 + (2.64 - 1.85i)T + (1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (0.958 - 3.57i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.03 - 1.78i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.25 - 1.78i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 0.543i)T + (10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-0.322 + 3.68i)T + (-22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (7.40 + 2.69i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.91 + 4.56i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.90 + 4.90i)T - 37iT^{2} \) |
| 41 | \( 1 + (7.67 + 1.35i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (9.90 - 0.866i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (4.11 + 8.82i)T + (-30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (9.05 + 0.791i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (1.99 - 0.726i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-8.62 + 7.23i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (2.31 - 1.07i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (2.53 - 3.01i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (3.03 + 4.33i)T + (-24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (-0.385 + 2.18i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-13.7 - 3.67i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.335 - 1.90i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.17 - 4.66i)T + (-62.3 - 74.3i)T^{2} \) |
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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
−10.54210275462873521606013378338, −9.796302713873608749764591058196, −9.470190059143770224853537556863, −8.278004132973891167513032590509, −6.63567782367496543078657064968, −5.94922384824532449663441875443, −5.24346720243390453918499480649, −4.50008041776766792640449465123, −3.51136145843895539457392368243, −2.07782456913405394285354285956,
0.01128679867455295199800105406, 1.20786823923670862228565922685, 3.17888586973948898750568665460, 4.64780430310949905266096862795, 5.27660084855557933263179333948, 6.28319011503531742766595459457, 6.85395236426291705844976024202, 7.51371109064677666898044227051, 8.203042128902442378950438647680, 9.765702765778631220879894461438