L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.606 + 0.350i)5-s + (−1.82 − 1.91i)7-s + 0.999i·8-s + (0.350 + 0.606i)10-s + (−2.78 + 1.80i)11-s − 7.03·13-s + (−0.629 − 2.56i)14-s + (−0.5 + 0.866i)16-s + (0.308 + 0.535i)17-s + (0.391 − 0.678i)19-s + 0.700i·20-s + (−3.31 + 0.176i)22-s + (−2.63 + 4.56i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.271 + 0.156i)5-s + (−0.691 − 0.722i)7-s + 0.353i·8-s + (0.110 + 0.191i)10-s + (−0.838 + 0.545i)11-s − 1.95·13-s + (−0.168 − 0.686i)14-s + (−0.125 + 0.216i)16-s + (0.0749 + 0.129i)17-s + (0.0898 − 0.155i)19-s + 0.156i·20-s + (−0.706 + 0.0375i)22-s + (−0.549 + 0.951i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4035057130\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4035057130\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.82 + 1.91i)T \) |
| 11 | \( 1 + (2.78 - 1.80i)T \) |
good | 5 | \( 1 + (-0.606 - 0.350i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 7.03T + 13T^{2} \) |
| 17 | \( 1 + (-0.308 - 0.535i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.391 + 0.678i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.63 - 4.56i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.09iT - 29T^{2} \) |
| 31 | \( 1 + (5.35 - 3.08i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.99 + 3.45i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.85T + 41T^{2} \) |
| 43 | \( 1 - 3.62iT - 43T^{2} \) |
| 47 | \( 1 + (-2.03 - 1.17i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.95 - 6.84i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.25 + 1.88i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.18 + 7.24i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.07 + 1.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + (-2.64 - 4.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.75 - 2.16i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.03T + 83T^{2} \) |
| 89 | \( 1 + (14.0 + 8.10i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−9.949797478893111005911444738935, −9.446988854107452434473946132144, −8.068359414983685630674715806938, −7.32169429023762418332496971653, −6.91389414032527062076875526467, −5.76668618801351810200010506327, −5.02062546692946079537273143691, −4.13799483297302827196244632367, −3.04725552771474292785368630188, −2.11674387007505209571021229323,
0.11640922216796685380384360132, 2.18284629139426638284913446539, 2.76295384691947562249926056497, 3.90889967601457631917596103689, 5.17395939236756318546265523707, 5.50034648850402893653179913982, 6.54507843288612919270698536115, 7.43096952200515548110830412721, 8.381430670667364263557093404361, 9.441138342221198036200501921409