L(s) = 1 | + (2.19 − 3.80i)5-s + 1.44·7-s − 15.1·11-s + (−14.8 + 8.57i)13-s + (−9.77 + 16.9i)17-s + (−3.34 + 18.7i)19-s + (2.19 + 3.80i)23-s + (2.84 + 4.93i)25-s + (29.3 − 16.9i)29-s + 5.62i·31-s + (3.18 − 5.51i)35-s + 23.7i·37-s + (32.2 + 18.6i)41-s + (−15.9 + 27.6i)43-s + (−39.0 − 67.7i)47-s + ⋯ |
L(s) = 1 | + (0.439 − 0.760i)5-s + 0.207·7-s − 1.37·11-s + (−1.14 + 0.659i)13-s + (−0.574 + 0.995i)17-s + (−0.176 + 0.984i)19-s + (0.0955 + 0.165i)23-s + (0.113 + 0.197i)25-s + (1.01 − 0.583i)29-s + 0.181i·31-s + (0.0909 − 0.157i)35-s + 0.641i·37-s + (0.787 + 0.454i)41-s + (−0.371 + 0.643i)43-s + (−0.831 − 1.44i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7856714316\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7856714316\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.34 - 18.7i)T \) |
good | 5 | \( 1 + (-2.19 + 3.80i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 - 1.44T + 49T^{2} \) |
| 11 | \( 1 + 15.1T + 121T^{2} \) |
| 13 | \( 1 + (14.8 - 8.57i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (9.77 - 16.9i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (-2.19 - 3.80i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-29.3 + 16.9i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 5.62iT - 961T^{2} \) |
| 37 | \( 1 - 23.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-32.2 - 18.6i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (15.9 - 27.6i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (39.0 + 67.7i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-9.55 + 5.51i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-78.4 - 45.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-57.1 - 98.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (98.3 - 56.7i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (87.9 + 50.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-10.0 + 17.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (45.8 + 26.4i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 29.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (126. - 73.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-27 - 15.5i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.27574407539795921881092111414, −9.892122453358532102324304240457, −8.700380294693476108106879904677, −8.118414624906917512068599658231, −7.09490578288472517132994039005, −5.94417466759543871805794636902, −5.07174143078094635642333679946, −4.29911801644164864028920385924, −2.69155936118757774017153476117, −1.58042429262620762102350260251,
0.26182243302771700753875927416, 2.43164709314542113442842171321, 2.88648179682267841225486944030, 4.70660896809465377779865730358, 5.31373069875138037310819141455, 6.58155358448382155553931567134, 7.31520956056582018157708389315, 8.152542862964103083344921668410, 9.272339720028452320396711806878, 10.14810546877935423782821594828