| L(s) = 1 | + (−1.03 + 0.966i)2-s + (−1.37 + 0.736i)3-s + (0.130 − 1.99i)4-s + (−0.634 − 0.773i)5-s + (0.710 − 2.09i)6-s + (−3.10 + 2.07i)7-s + (1.79 + 2.18i)8-s + (−0.308 + 0.462i)9-s + (1.40 + 0.184i)10-s + (−2.82 + 0.855i)11-s + (1.29 + 2.84i)12-s + (2.98 + 2.44i)13-s + (1.19 − 5.14i)14-s + (1.44 + 0.598i)15-s + (−3.96 − 0.520i)16-s + (0.554 − 0.229i)17-s + ⋯ |
| L(s) = 1 | + (−0.729 + 0.683i)2-s + (−0.796 + 0.425i)3-s + (0.0651 − 0.997i)4-s + (−0.283 − 0.345i)5-s + (0.290 − 0.854i)6-s + (−1.17 + 0.783i)7-s + (0.634 + 0.772i)8-s + (−0.102 + 0.154i)9-s + (0.443 + 0.0583i)10-s + (−0.850 + 0.258i)11-s + (0.372 + 0.822i)12-s + (0.827 + 0.679i)13-s + (0.320 − 1.37i)14-s + (0.372 + 0.154i)15-s + (−0.991 − 0.130i)16-s + (0.134 − 0.0556i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.185029 - 0.0855813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185029 - 0.0855813i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.03 - 0.966i)T \) |
| 5 | \( 1 + (0.634 + 0.773i)T \) |
| good | 3 | \( 1 + (1.37 - 0.736i)T + (1.66 - 2.49i)T^{2} \) |
| 7 | \( 1 + (3.10 - 2.07i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (2.82 - 0.855i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-2.98 - 2.44i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.554 + 0.229i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.0238 - 0.241i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-1.30 + 0.258i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-2.17 + 7.18i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (7.25 + 7.25i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.61 - 0.158i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.48 + 7.48i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (4.15 + 2.22i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-1.45 - 3.50i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.554 - 1.82i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-11.5 + 9.49i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-4.17 - 7.81i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-3.16 - 5.91i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-0.0539 - 0.0806i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (5.76 + 3.85i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.358 - 0.866i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (1.07 - 0.106i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-7.59 - 1.51i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (11.9 + 11.9i)T + 97iT^{2} \) |
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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
−10.26530110191617816306051832140, −9.573039198753091022748594639545, −8.770950469181663263990646958111, −7.907541156192584573712124293832, −6.79136001119576015367363451061, −5.86467517573646383412614460305, −5.38382052879579282631590933693, −4.13650060575244662138113129110, −2.35096350462576841759516690895, −0.18970066234425690268094871913,
1.03632506092690888223591252990, 3.06448041459619926785611799323, 3.60732347897278078569735184644, 5.31841963985140266782180142864, 6.57026005174270565099755019916, 7.06354481458881377634846506682, 8.118693811884191787755436402509, 9.046445091532758224404947898337, 10.16308140241981153840585386391, 10.68140173363958492330274834497
