| L(s) = 1 | + (−0.603 + 1.04i)2-s + (−1.82 − 1.52i)3-s + (0.272 + 0.472i)4-s + (−0.272 + 1.54i)5-s + (2.69 − 0.980i)6-s + (−0.703 + 3.98i)7-s − 3.06·8-s + (0.459 + 2.60i)9-s + (−1.44 − 1.21i)10-s + (0.248 − 0.208i)11-s + (0.224 − 1.27i)12-s + (−0.817 + 4.63i)13-s + (−3.74 − 3.13i)14-s + (2.85 − 2.39i)15-s + (1.30 − 2.26i)16-s + (2.72 − 4.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.426 + 0.738i)2-s + (−1.05 − 0.882i)3-s + (0.136 + 0.236i)4-s + (−0.121 + 0.690i)5-s + (1.09 − 0.400i)6-s + (−0.265 + 1.50i)7-s − 1.08·8-s + (0.153 + 0.869i)9-s + (−0.457 − 0.384i)10-s + (0.0750 − 0.0629i)11-s + (0.0649 − 0.368i)12-s + (−0.226 + 1.28i)13-s + (−1.00 − 0.839i)14-s + (0.736 − 0.618i)15-s + (0.326 − 0.565i)16-s + (0.661 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.240055 + 0.464438i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.240055 + 0.464438i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (4.49 + 9.42i)T \) |
| good | 2 | \( 1 + (0.603 - 1.04i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.82 + 1.52i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (0.272 - 1.54i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.703 - 3.98i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.248 + 0.208i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.817 - 4.63i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.72 + 4.72i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.155 + 0.268i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.60 + 2.78i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.67 - 5.59i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.06 + 6.01i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.65 - 9.37i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 + (-3.13 + 5.42i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.15 + 1.51i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (0.836 - 4.74i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (0.556 - 0.466i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 3.77i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.02 - 11.5i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.27 - 10.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.64 + 3.89i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-0.811 - 4.60i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.01 - 5.88i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (12.1 - 4.42i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.78 + 15.8i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−14.23033728720965745129224522758, −12.60678541152887793544233478918, −11.91092315001416873410664183473, −11.41057473954819212090056990924, −9.550760446281399690158668906047, −8.427539398626964279723267929971, −6.99416106847449743383063190381, −6.56036655256468938263935795408, −5.43035354772089818339376702921, −2.71252632151486348745426390815,
0.77238631182607221033595561999, 3.70452201675123396429848632728, 5.08007826361879945289119367020, 6.26282115079862078896566753196, 8.006197496067680907250463674893, 9.656084705705820288585497967944, 10.40777252134282629035412274607, 10.79897009180006194843432939458, 12.04514925224058023422872628540, 12.92176906550421798306905531945
