| L(s) = 1 | + 1.02·3-s + i·5-s + (1.49 − 2.18i)7-s − 1.95·9-s − 2.58i·11-s − 5.79i·13-s + 1.02i·15-s + 4.41i·17-s − 3.62·19-s + (1.52 − 2.22i)21-s − 6.61i·23-s − 25-s − 5.06·27-s − 10.1·29-s − 1.36·31-s + ⋯ |
| L(s) = 1 | + 0.589·3-s + 0.447i·5-s + (0.564 − 0.825i)7-s − 0.652·9-s − 0.778i·11-s − 1.60i·13-s + 0.263i·15-s + 1.07i·17-s − 0.831·19-s + (0.332 − 0.486i)21-s − 1.37i·23-s − 0.200·25-s − 0.974·27-s − 1.87·29-s − 0.245·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.269108876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.269108876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-1.49 + 2.18i)T \) |
| good | 3 | \( 1 - 1.02T + 3T^{2} \) |
| 11 | \( 1 + 2.58iT - 11T^{2} \) |
| 13 | \( 1 + 5.79iT - 13T^{2} \) |
| 17 | \( 1 - 4.41iT - 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 + 6.61iT - 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 - 2.41iT - 41T^{2} \) |
| 43 | \( 1 - 8.71iT - 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 9.00iT - 61T^{2} \) |
| 67 | \( 1 + 8.04iT - 67T^{2} \) |
| 71 | \( 1 - 6.09iT - 71T^{2} \) |
| 73 | \( 1 + 5.74iT - 73T^{2} \) |
| 79 | \( 1 - 6.97iT - 79T^{2} \) |
| 83 | \( 1 - 8.92T + 83T^{2} \) |
| 89 | \( 1 + 0.519iT - 89T^{2} \) |
| 97 | \( 1 + 9.08iT - 97T^{2} \) |
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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
−8.507035289668641566263173684310, −8.059675843905999799286391539199, −7.53002477077948162444932740729, −6.28374015873534762301971091356, −5.79744290149826942985870517504, −4.64580912939276210806902719028, −3.63285073018802002990319208120, −3.02653141108151811553806098298, −1.90504491288565638104949264795, −0.36772112278795502008417065597,
1.80501008672994692658292995462, 2.28301009858153057156104063498, 3.58754465962063104889093318664, 4.49802555142734359621982656416, 5.30797896990664787804951122019, 6.05485907513886854587627923591, 7.25030932455409059241941563190, 7.71296036667104892434660938362, 8.904910960580048539138940770774, 9.042557782550397687137701529587
