| L(s) = 1 | + (0.947 − 1.04i)2-s + (−0.204 − 1.98i)4-s + i·5-s + (−2.29 − 1.31i)7-s + (−2.28 − 1.66i)8-s + (1.04 + 0.947i)10-s − 0.477i·11-s + 2.96i·13-s + (−3.55 + 1.16i)14-s + (−3.91 + 0.814i)16-s − 3.83i·17-s − 5.31·19-s + (1.98 − 0.204i)20-s + (−0.500 − 0.452i)22-s − 7.60i·23-s + ⋯ |
| L(s) = 1 | + (0.669 − 0.742i)2-s + (−0.102 − 0.994i)4-s + 0.447i·5-s + (−0.868 − 0.496i)7-s + (−0.807 − 0.590i)8-s + (0.332 + 0.299i)10-s − 0.143i·11-s + 0.821i·13-s + (−0.950 + 0.311i)14-s + (−0.979 + 0.203i)16-s − 0.929i·17-s − 1.21·19-s + (0.444 − 0.0457i)20-s + (−0.106 − 0.0963i)22-s − 1.58i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7951188969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7951188969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.947 + 1.04i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.29 + 1.31i)T \) |
| good | 11 | \( 1 + 0.477iT - 11T^{2} \) |
| 13 | \( 1 - 2.96iT - 13T^{2} \) |
| 17 | \( 1 + 3.83iT - 17T^{2} \) |
| 19 | \( 1 + 5.31T + 19T^{2} \) |
| 23 | \( 1 + 7.60iT - 23T^{2} \) |
| 29 | \( 1 + 6.17T + 29T^{2} \) |
| 31 | \( 1 + 3.38T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 - 1.01iT - 41T^{2} \) |
| 43 | \( 1 - 6.85iT - 43T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 - 9.30T + 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 + 4.75iT - 61T^{2} \) |
| 67 | \( 1 + 1.30iT - 67T^{2} \) |
| 71 | \( 1 + 9.18iT - 71T^{2} \) |
| 73 | \( 1 - 4.49iT - 73T^{2} \) |
| 79 | \( 1 + 8.80iT - 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 1.60iT - 97T^{2} \) |
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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
−9.385685785436594357381051156088, −8.756901264961865497952206857667, −7.24867130600110794040237805308, −6.62447595216805361956624619085, −5.89987890301830376852792059756, −4.67311337712612272963180218290, −3.92524687926565706982665119524, −2.98636156482953676338930798073, −2.00051272489352150894579984350, −0.24435487051183644983169442760,
2.12547137246431547586054220519, 3.44116634504635294245187671578, 4.06085877665844860221397878956, 5.53098267087325937753597736534, 5.65538907793208255988719815845, 6.80686637309102907690515929106, 7.55061256560996120034474078585, 8.551287087715822835126893092955, 9.020216467839851276066571898969, 10.03756342207474253860491441397
