L(s) = 1 | − 1.33·5-s + (2.54 − 0.728i)7-s + 1.51·11-s + (−2.58 + 4.48i)13-s + (−0.774 + 1.34i)17-s + (1.25 + 2.16i)19-s − 7.36·23-s − 3.21·25-s + (0.0309 + 0.0536i)29-s + (−1.92 − 3.33i)31-s + (−3.39 + 0.972i)35-s + (−0.281 − 0.487i)37-s + (−4.51 + 7.81i)41-s + (−5.09 − 8.83i)43-s + (4.75 − 8.24i)47-s + ⋯ |
L(s) = 1 | − 0.596·5-s + (0.961 − 0.275i)7-s + 0.456·11-s + (−0.717 + 1.24i)13-s + (−0.187 + 0.325i)17-s + (0.287 + 0.497i)19-s − 1.53·23-s − 0.643·25-s + (0.00575 + 0.00996i)29-s + (−0.345 − 0.598i)31-s + (−0.573 + 0.164i)35-s + (−0.0462 − 0.0801i)37-s + (−0.704 + 1.22i)41-s + (−0.777 − 1.34i)43-s + (0.694 − 1.20i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5654189635\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5654189635\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.54 + 0.728i)T \) |
good | 5 | \( 1 + 1.33T + 5T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 + (2.58 - 4.48i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.774 - 1.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 2.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7.36T + 23T^{2} \) |
| 29 | \( 1 + (-0.0309 - 0.0536i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.92 + 3.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.51 - 7.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.75 + 8.24i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.755 - 1.30i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.22 - 7.31i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.61 - 2.80i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 6.00i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.95 - 5.12i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.80 - 4.85i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.703 + 1.21i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.09 + 10.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.874063629709997702774468999634, −8.257928620381421478382914959645, −7.53885972869489807296000504481, −6.95684551240009033533319047436, −5.99608119532253908987000611556, −5.10932273054964173256011656056, −4.07466105765457491681835242621, −3.92904536400960711884664631292, −2.28876544850719587946546315293, −1.51531621847283679693906226924,
0.17091861064912731097742151325, 1.61220166574468911965902281965, 2.67876008326401883077781134292, 3.66898634168671207133253080841, 4.57167350537345144649726951369, 5.24982907480455896767170793983, 6.04471261539503732073291885954, 7.09491262433252997765057915027, 7.83074574224125481603582768945, 8.182758337634850353069044822859