| L(s) = 1 | + (−1.10 − 0.881i)2-s + (0.446 + 1.94i)4-s + 0.946i·5-s − 1.89i·7-s + (1.22 − 2.54i)8-s + (0.834 − 1.04i)10-s − 4.64·11-s + 4.52·13-s + (−1.67 + 2.09i)14-s + (−3.60 + 1.73i)16-s − 4.07·17-s + (−3.46 + 2.64i)19-s + (−1.84 + 0.422i)20-s + (5.13 + 4.08i)22-s + 5.34i·23-s + ⋯ |
| L(s) = 1 | + (−0.782 − 0.623i)2-s + (0.223 + 0.974i)4-s + 0.423i·5-s − 0.717i·7-s + (0.433 − 0.901i)8-s + (0.263 − 0.331i)10-s − 1.39·11-s + 1.25·13-s + (−0.447 + 0.561i)14-s + (−0.900 + 0.434i)16-s − 0.989·17-s + (−0.794 + 0.607i)19-s + (−0.412 + 0.0944i)20-s + (1.09 + 0.871i)22-s + 1.11i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4139831897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4139831897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.10 + 0.881i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.46 - 2.64i)T \) |
| good | 5 | \( 1 - 0.946iT - 5T^{2} \) |
| 7 | \( 1 + 1.89iT - 7T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + 4.07T + 17T^{2} \) |
| 23 | \( 1 - 5.34iT - 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 11.0iT - 41T^{2} \) |
| 43 | \( 1 + 7.20T + 43T^{2} \) |
| 47 | \( 1 - 8.10iT - 47T^{2} \) |
| 53 | \( 1 + 4.02T + 53T^{2} \) |
| 59 | \( 1 - 9.43iT - 59T^{2} \) |
| 61 | \( 1 - 15.2iT - 61T^{2} \) |
| 67 | \( 1 - 9.68iT - 67T^{2} \) |
| 71 | \( 1 + 3.78T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 6.42T + 79T^{2} \) |
| 83 | \( 1 + 0.0483T + 83T^{2} \) |
| 89 | \( 1 - 3.55iT - 89T^{2} \) |
| 97 | \( 1 + 2.28iT - 97T^{2} \) |
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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
−10.24739366447595120943049413211, −8.784325514975301344247299571839, −8.514836773303643464995646911577, −7.41750934201671847758272795610, −6.90174552241496662284749351022, −5.79095654074391396815174250003, −4.45506665487858471863492929350, −3.55437278156152353511383697760, −2.62982520857487353174364180510, −1.39910757418311197821861158858,
0.21987211196169476994175777838, 1.83885674261263366679111795815, 2.90582326055678439455778582389, 4.68634711778284393063681622400, 5.18311689250499647561297422751, 6.33966318344013578847089762408, 6.74982508046759255092407023716, 8.124481112650009008348008080851, 8.539760976764100945296065614403, 8.977435793824112053277193994746
