| L(s) = 1 | + (1.23 − 0.684i)2-s + (−0.404 − 0.526i)3-s + (1.06 − 1.69i)4-s + (−1.31 − 1.00i)5-s + (−0.860 − 0.375i)6-s + (−2.64 − 0.158i)7-s + (0.158 − 2.82i)8-s + (0.662 − 2.47i)9-s + (−2.31 − 0.348i)10-s + (4.60 + 0.605i)11-s + (−1.32 + 0.124i)12-s + (0.221 + 0.0917i)13-s + (−3.37 + 1.61i)14-s + 1.09i·15-s + (−1.73 − 3.60i)16-s + (3.43 + 1.98i)17-s + ⋯ |
| L(s) = 1 | + (0.875 − 0.483i)2-s + (−0.233 − 0.303i)3-s + (0.531 − 0.846i)4-s + (−0.586 − 0.450i)5-s + (−0.351 − 0.153i)6-s + (−0.998 − 0.0600i)7-s + (0.0559 − 0.998i)8-s + (0.220 − 0.824i)9-s + (−0.731 − 0.110i)10-s + (1.38 + 0.182i)11-s + (−0.381 + 0.0358i)12-s + (0.0614 + 0.0254i)13-s + (−0.902 + 0.430i)14-s + 0.283i·15-s + (−0.434 − 0.900i)16-s + (0.832 + 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06400 - 1.23848i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06400 - 1.23848i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 0.684i)T \) |
| 7 | \( 1 + (2.64 + 0.158i)T \) |
| good | 3 | \( 1 + (0.404 + 0.526i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (1.31 + 1.00i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (-4.60 - 0.605i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (-0.221 - 0.0917i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (-3.43 - 1.98i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.846 - 6.43i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (1.61 - 6.02i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.351 + 0.848i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-4.24 + 7.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.66 - 5.11i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (5.90 + 5.90i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.33 - 5.63i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (6.98 - 4.03i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.96 + 0.389i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (1.14 - 8.67i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-4.94 + 0.651i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (3.05 + 3.98i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-4.04 + 4.04i)T - 71iT^{2} \) |
| 73 | \( 1 + (-10.3 + 2.77i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.660 - 0.381i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.95 + 0.808i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (8.29 + 2.22i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 - 7.03T + 97T^{2} \) |
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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
−12.05950669834701783944608153657, −11.60389501761479974431088644061, −9.963114478307164034332539509614, −9.490827069902854255934140211759, −7.80153779490230048590994380864, −6.47840380744805838316458989895, −5.92210479006506549676715225527, −4.11765512968910429319088539065, −3.50946075344481337242761920134, −1.25226454304339714354984584673,
2.91153215192128931455146621692, 3.99069005142794826090731285471, 5.13882895463934857708814353620, 6.51008708134346538817228761918, 7.09665217022239465010190348411, 8.379771951282481231506434482627, 9.636920253507014547299504929267, 10.90374525563860179603392800894, 11.66627622073419510337855292026, 12.54068804781182468895801847340
