| L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.0710 + 1.73i)3-s + (−0.978 − 0.207i)4-s − 0.0318·5-s + (1.71 + 0.251i)6-s + (1.17 + 3.60i)7-s + (−0.309 + 0.951i)8-s + (−2.98 − 0.245i)9-s + (−0.00332 + 0.0316i)10-s + (−5.31 − 1.12i)11-s + (0.429 − 1.67i)12-s + (−1.68 + 1.22i)13-s + (3.71 − 0.788i)14-s + (0.00226 − 0.0551i)15-s + (0.913 + 0.406i)16-s + (0.707 + 0.785i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (−0.0410 + 0.999i)3-s + (−0.489 − 0.103i)4-s − 0.0142·5-s + (0.699 + 0.102i)6-s + (0.443 + 1.36i)7-s + (−0.109 + 0.336i)8-s + (−0.996 − 0.0819i)9-s + (−0.00105 + 0.0100i)10-s + (−1.60 − 0.340i)11-s + (0.123 − 0.484i)12-s + (−0.466 + 0.338i)13-s + (0.991 − 0.210i)14-s + (0.000584 − 0.0142i)15-s + (0.228 + 0.101i)16-s + (0.171 + 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.372503 + 0.678832i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.372503 + 0.678832i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 3 | \( 1 + (0.0710 - 1.73i)T \) |
| 31 | \( 1 + (-5.37 + 1.46i)T \) |
| good | 5 | \( 1 + 0.0318T + 5T^{2} \) |
| 7 | \( 1 + (-1.17 - 3.60i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (5.31 + 1.12i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.68 - 1.22i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.707 - 0.785i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 0.461i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (4.33 + 4.81i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.848 - 8.07i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (5.24 - 9.08i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.25 - 4.54i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-7.73 - 5.62i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (7.55 + 3.36i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (6.74 - 1.43i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-9.47 - 4.21i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-2.87 - 4.97i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + (-7.58 + 1.61i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-2.39 + 2.66i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (0.571 - 1.75i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-12.1 + 5.39i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (1.97 + 6.07i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (12.8 - 14.3i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−11.01894404221373348596172972269, −10.22397710414738743008654162127, −9.532350933771914255246516427375, −8.526917880202924680869008462769, −8.011365047806294454160700317391, −6.07087390376555950850616029169, −5.24799817948830795518078574969, −4.57187644810630834123940029900, −3.09418632501202532198052826105, −2.31177792844643178818317142179,
0.41014864607115496585968748918, 2.23123710922009041777663851569, 3.83005846068349509535831293904, 5.10643090033267086738709681264, 5.90393741522293825972516789652, 7.16469467128168396885049226031, 7.76962471687479547639001140271, 8.012472567916107113655529447504, 9.621829423790761212169828815964, 10.42400333258762184491585977959
