| L(s) = 1 | + (0.949 + 1.04i)2-s + (−1.24 + 1.19i)3-s + (−0.196 + 1.99i)4-s + (0.101 + 2.23i)5-s + (−2.44 − 0.169i)6-s + 0.903·7-s + (−2.27 + 1.68i)8-s + (0.120 − 2.99i)9-s + (−2.24 + 2.22i)10-s + (0.0159 + 0.0115i)11-s + (−2.14 − 2.72i)12-s + (−1.50 − 2.07i)13-s + (0.857 + 0.946i)14-s + (−2.80 − 2.66i)15-s + (−3.92 − 0.783i)16-s + (0.0389 − 0.119i)17-s + ⋯ |
| L(s) = 1 | + (0.671 + 0.741i)2-s + (−0.721 + 0.692i)3-s + (−0.0983 + 0.995i)4-s + (0.0452 + 0.998i)5-s + (−0.997 − 0.0692i)6-s + 0.341·7-s + (−0.803 + 0.595i)8-s + (0.0401 − 0.999i)9-s + (−0.709 + 0.704i)10-s + (0.00480 + 0.00349i)11-s + (−0.618 − 0.785i)12-s + (−0.417 − 0.574i)13-s + (0.229 + 0.253i)14-s + (−0.724 − 0.689i)15-s + (−0.980 − 0.195i)16-s + (0.00943 − 0.0290i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.209997 + 1.34029i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.209997 + 1.34029i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.949 - 1.04i)T \) |
| 3 | \( 1 + (1.24 - 1.19i)T \) |
| 5 | \( 1 + (-0.101 - 2.23i)T \) |
| good | 7 | \( 1 - 0.903T + 7T^{2} \) |
| 11 | \( 1 + (-0.0159 - 0.0115i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.50 + 2.07i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.0389 + 0.119i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.78 - 1.87i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (4.34 - 5.98i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.74 + 2.19i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.78 + 0.579i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.02 - 2.78i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.58 - 3.55i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 + (-5.03 + 1.63i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.0726 + 0.223i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-12.0 + 8.78i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.52 - 4.01i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (4.13 - 12.7i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.59 + 11.0i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.17 - 4.36i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.586 - 0.190i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (11.0 + 3.58i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.84 + 8.04i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-12.3 + 4.00i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.87249754722508651934525499411, −11.55119172522062636165736758229, −10.31310833122751102154251823534, −9.524309221103489361390644365115, −8.001363864192744371070161627464, −7.16061606734034729293208175351, −6.03358886696886074378553235302, −5.33806527754363055803668904434, −4.08080613664804315129762621485, −3.00536927907864056955912408490,
0.935730019154633453949988083172, 2.31683204886871244272922444207, 4.32434094121040545846009184046, 5.11967766342283829710466803627, 6.04419621558849857919062040673, 7.25545645025831968096710239003, 8.553133555116109620811648619365, 9.651146325998078785480822690800, 10.68176088128199893235559576456, 11.67526632926009897347232045450
