| L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.0437 − 0.304i)3-s + (0.841 − 0.540i)4-s + (−0.415 + 0.909i)5-s + (0.127 + 0.279i)6-s + (−4.20 + 1.23i)7-s + (−0.654 + 0.755i)8-s + (2.78 − 0.818i)9-s + (0.142 − 0.989i)10-s + (1.37 − 3.00i)11-s + (−0.201 − 0.232i)12-s + (−2.09 − 2.42i)13-s + (3.68 − 2.36i)14-s + (0.295 + 0.0866i)15-s + (0.415 − 0.909i)16-s + (2.96 + 1.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (−0.0252 − 0.175i)3-s + (0.420 − 0.270i)4-s + (−0.185 + 0.406i)5-s + (0.0521 + 0.114i)6-s + (−1.58 + 0.466i)7-s + (−0.231 + 0.267i)8-s + (0.929 − 0.272i)9-s + (0.0450 − 0.313i)10-s + (0.413 − 0.904i)11-s + (−0.0581 − 0.0671i)12-s + (−0.581 − 0.671i)13-s + (0.985 − 0.633i)14-s + (0.0762 + 0.0223i)15-s + (0.103 − 0.227i)16-s + (0.719 + 0.462i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.921360 + 0.205783i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921360 + 0.205783i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (-7.98 + 1.78i)T \) |
| good | 3 | \( 1 + (0.0437 + 0.304i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (4.20 - 1.23i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.37 + 3.00i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (2.09 + 2.42i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.96 - 1.90i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-7.14 - 2.09i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (-0.750 - 5.22i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 + (5.12 - 5.91i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + (-0.690 - 0.443i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-9.49 - 6.09i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (1.18 + 8.26i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (-3.52 + 2.26i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (6.45 - 7.44i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.230 + 0.504i)T + (-39.9 + 46.1i)T^{2} \) |
| 71 | \( 1 + (4.43 - 2.85i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.260 + 0.570i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.90 - 2.19i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (2.16 - 4.74i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-1.45 + 10.1i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−10.18646144594023504175697964459, −9.755927128801682417743207579499, −9.046521204417823215165710122336, −7.77221682426274569208294918264, −7.18039801344888731559002583637, −6.20832305277047060931656457688, −5.55435674632296871250408533416, −3.61611983768988088416823478043, −2.94755751169572650457004189145, −1.02173543521183554922097545460,
0.866218618379938445472018380986, 2.56445219672600289468062912707, 3.83405498872046435826329529323, 4.74057568778664052464782028408, 6.23184718359992881816241732408, 7.27951405058762521326743074782, 7.50599018761412005278989486133, 9.239694657809996872036477948425, 9.561954062363619588681171981755, 10.07141448600146944521085461758
