| L(s) = 1 | + (1.34 − 0.439i)2-s + (0.0625 − 0.0755i)3-s + (1.61 − 1.18i)4-s + (2.10 − 0.743i)5-s + (0.0508 − 0.129i)6-s + (0.464 + 0.150i)7-s + (1.65 − 2.29i)8-s + (0.560 + 2.93i)9-s + (2.50 − 1.92i)10-s + (0.237 + 1.88i)11-s + (0.0116 − 0.195i)12-s + (0.985 + 5.16i)13-s + (0.690 − 0.00123i)14-s + (0.0757 − 0.205i)15-s + (1.20 − 3.81i)16-s + (−0.807 + 3.14i)17-s + ⋯ |
| L(s) = 1 | + (0.950 − 0.310i)2-s + (0.0361 − 0.0436i)3-s + (0.806 − 0.590i)4-s + (0.943 − 0.332i)5-s + (0.0207 − 0.0527i)6-s + (0.175 + 0.0570i)7-s + (0.583 − 0.812i)8-s + (0.186 + 0.979i)9-s + (0.793 − 0.608i)10-s + (0.0717 + 0.567i)11-s + (0.00335 − 0.0565i)12-s + (0.273 + 1.43i)13-s + (0.184 − 0.000330i)14-s + (0.0195 − 0.0531i)15-s + (0.302 − 0.953i)16-s + (−0.195 + 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.49642 - 0.705630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.49642 - 0.705630i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.34 + 0.439i)T \) |
| 5 | \( 1 + (-2.10 + 0.743i)T \) |
| good | 3 | \( 1 + (-0.0625 + 0.0755i)T + (-0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (-0.464 - 0.150i)T + (5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.237 - 1.88i)T + (-10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (-0.985 - 5.16i)T + (-12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (0.807 - 3.14i)T + (-14.8 - 8.18i)T^{2} \) |
| 19 | \( 1 + (-3.68 + 3.05i)T + (3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (4.54 + 4.84i)T + (-1.44 + 22.9i)T^{2} \) |
| 29 | \( 1 + (5.64 + 0.355i)T + (28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (-2.67 - 0.686i)T + (27.1 + 14.9i)T^{2} \) |
| 37 | \( 1 + (5.77 - 3.17i)T + (19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (8.44 + 7.93i)T + (2.57 + 40.9i)T^{2} \) |
| 43 | \( 1 + (4.06 + 2.95i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-3.01 + 7.62i)T + (-34.2 - 32.1i)T^{2} \) |
| 53 | \( 1 + (1.79 + 2.83i)T + (-22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-0.228 - 0.107i)T + (37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (2.92 + 3.11i)T + (-3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.642 - 10.2i)T + (-66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (2.70 + 1.06i)T + (51.7 + 48.6i)T^{2} \) |
| 73 | \( 1 + (-1.99 + 0.939i)T + (46.5 - 56.2i)T^{2} \) |
| 79 | \( 1 + (-10.8 + 13.0i)T + (-14.8 - 77.6i)T^{2} \) |
| 83 | \( 1 + (-10.0 - 12.1i)T + (-15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (-5.72 - 12.1i)T + (-56.7 + 68.5i)T^{2} \) |
| 97 | \( 1 + (0.488 + 0.0307i)T + (96.2 + 12.1i)T^{2} \) |
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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
−10.15674157883469409435797650145, −9.262418143040766093977066534340, −8.295900671969595283924422684543, −7.01810160582940848043676957490, −6.47684900163886334286353513595, −5.32242480647156433667002812947, −4.78263905719872397346575025211, −3.78013088768507205827260361600, −2.15670598354214714201514477732, −1.77468636998950885820529470025,
1.51206240813572795473644694014, 3.05143402429179461862995320877, 3.53971394537187361121531890574, 5.00159714374230223283714033053, 5.80051786634631685126012350084, 6.30245719601228843720768672172, 7.38340038309913420660099168308, 8.116770255370538815934057995832, 9.318703805842834249482030110118, 10.04676484177953071114452350383
