| L(s) = 1 | + (1.39 − 0.208i)2-s + (1.21 − 2.60i)3-s + (1.91 − 0.582i)4-s + (0.203 + 2.22i)5-s + (1.15 − 3.89i)6-s + (−0.256 + 0.957i)7-s + (2.55 − 1.21i)8-s + (−3.36 − 4.01i)9-s + (0.748 + 3.07i)10-s + (−0.697 − 0.402i)11-s + (0.806 − 5.68i)12-s + (0.756 + 1.62i)13-s + (−0.159 + 1.39i)14-s + (6.03 + 2.17i)15-s + (3.32 − 2.22i)16-s + (−5.90 + 0.516i)17-s + ⋯ |
| L(s) = 1 | + (0.989 − 0.147i)2-s + (0.700 − 1.50i)3-s + (0.956 − 0.291i)4-s + (0.0912 + 0.995i)5-s + (0.471 − 1.58i)6-s + (−0.0969 + 0.361i)7-s + (0.903 − 0.428i)8-s + (−1.12 − 1.33i)9-s + (0.236 + 0.971i)10-s + (−0.210 − 0.121i)11-s + (0.232 − 1.64i)12-s + (0.209 + 0.449i)13-s + (−0.0426 + 0.372i)14-s + (1.55 + 0.560i)15-s + (0.830 − 0.557i)16-s + (−1.43 + 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.51259 - 1.45247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.51259 - 1.45247i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.208i)T \) |
| 5 | \( 1 + (-0.203 - 2.22i)T \) |
| 19 | \( 1 + (4.35 + 0.104i)T \) |
| good | 3 | \( 1 + (-1.21 + 2.60i)T + (-1.92 - 2.29i)T^{2} \) |
| 7 | \( 1 + (0.256 - 0.957i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.697 + 0.402i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.756 - 1.62i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (5.90 - 0.516i)T + (16.7 - 2.95i)T^{2} \) |
| 23 | \( 1 + (-4.92 - 3.44i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.865 + 1.03i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (5.31 - 3.06i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.656 + 0.656i)T - 37iT^{2} \) |
| 41 | \( 1 + (-9.74 + 3.54i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (4.73 + 6.76i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (11.4 + 1.00i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (-2.03 + 2.90i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-8.64 - 7.25i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.0190 + 0.107i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.21 - 0.368i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-12.7 - 2.25i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.94 - 1.84i)T + (46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (1.15 - 0.420i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.83 - 1.02i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (1.52 - 4.18i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (9.06 - 0.793i)T + (95.5 - 16.8i)T^{2} \) |
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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
−11.35351331064615225267124900043, −10.76946121583607989876838304449, −9.227830996168554453874243527405, −8.140433561593244132179615778342, −6.90240474101899490077545613999, −6.78546135582270622062265770698, −5.61449601570601629452338017408, −3.84538237454285577794969145908, −2.64659576506847694544021690693, −1.93449997835353979482012108991,
2.42399585572874491219213053415, 3.75657642655654714410147483052, 4.52808411986634876675574726416, 5.17046157195405859225091362178, 6.50615091512807784952845619239, 7.994008258775865226013488250298, 8.771043054319975805035117965465, 9.677228392265164296444596064922, 10.74006087348455895779837974366, 11.27890466780301859796438976668
