L(s) = 1 | + i·2-s − 4-s + 0.266i·5-s + (−1.34 − 2.27i)7-s − i·8-s − 0.266·10-s + (−1.62 + 2.89i)11-s + 2.55·13-s + (2.27 − 1.34i)14-s + 16-s + 4.96·17-s − 8.21·19-s − 0.266i·20-s + (−2.89 − 1.62i)22-s + 5.23·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.118i·5-s + (−0.509 − 0.860i)7-s − 0.353i·8-s − 0.0841·10-s + (−0.489 + 0.871i)11-s + 0.707·13-s + (0.608 − 0.360i)14-s + 0.250·16-s + 1.20·17-s − 1.88·19-s − 0.0594i·20-s + (−0.616 − 0.346i)22-s + 1.09·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.347519432\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.347519432\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.34 + 2.27i)T \) |
| 11 | \( 1 + (1.62 - 2.89i)T \) |
good | 5 | \( 1 - 0.266iT - 5T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 - 5.25iT - 29T^{2} \) |
| 31 | \( 1 - 4.28iT - 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 + 6.21T + 41T^{2} \) |
| 43 | \( 1 + 1.46iT - 43T^{2} \) |
| 47 | \( 1 - 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 9.10iT - 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.93T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 - 6.16iT - 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 1.78iT - 89T^{2} \) |
| 97 | \( 1 + 0.146iT - 97T^{2} \) |
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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
−9.786998060405489556092680289908, −8.814202516952155555188077946183, −8.134845304295891980507560912786, −7.07749648318206365182296300300, −6.79835164178119195644142286985, −5.72307694664673083719129475598, −4.75645492548475997268043059182, −3.92583394437717635241497649678, −2.86059656683683872351476360127, −1.12845930899476964938928448555,
0.66236665493023983087966835817, 2.23411306119386551427502474238, 3.10241737984337552151508876162, 4.00958612522419509390573386076, 5.22963131284920749960299203636, 5.91055757170386449439546755215, 6.77525345473457783706740015889, 8.281052958505975349670542640627, 8.477973024907906035802468858542, 9.427453471394159860192040845293