| L(s) = 1 | + (1.00 − 0.993i)2-s + (−0.376 + 1.89i)3-s + (0.0253 − 1.99i)4-s + (1.44 − 2.16i)5-s + (1.50 + 2.27i)6-s + (−2.16 − 1.52i)7-s + (−1.96 − 2.03i)8-s + (−0.665 − 0.275i)9-s + (−0.694 − 3.60i)10-s + (−0.739 − 3.71i)11-s + (3.77 + 0.800i)12-s + (−0.636 − 0.952i)13-s + (−3.69 + 0.618i)14-s + (3.54 + 3.54i)15-s + (−3.99 − 0.101i)16-s + (2.60 − 2.60i)17-s + ⋯ |
| L(s) = 1 | + (0.711 − 0.702i)2-s + (−0.217 + 1.09i)3-s + (0.0126 − 0.999i)4-s + (0.645 − 0.966i)5-s + (0.612 + 0.929i)6-s + (−0.817 − 0.575i)7-s + (−0.693 − 0.720i)8-s + (−0.221 − 0.0919i)9-s + (−0.219 − 1.14i)10-s + (−0.222 − 1.12i)11-s + (1.08 + 0.231i)12-s + (−0.176 − 0.264i)13-s + (−0.986 + 0.165i)14-s + (0.915 + 0.915i)15-s + (−0.999 − 0.0253i)16-s + (0.631 − 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0360 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0360 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33186 - 1.28470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33186 - 1.28470i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.00 + 0.993i)T \) |
| 7 | \( 1 + (2.16 + 1.52i)T \) |
| good | 3 | \( 1 + (0.376 - 1.89i)T + (-2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (-1.44 + 2.16i)T + (-1.91 - 4.61i)T^{2} \) |
| 11 | \( 1 + (0.739 + 3.71i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (0.636 + 0.952i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-2.60 + 2.60i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.14 - 3.21i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-3.20 - 1.32i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-7.43 - 1.47i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 0.278iT - 31T^{2} \) |
| 37 | \( 1 + (-1.27 + 1.90i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (9.87 + 4.09i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (4.62 - 0.919i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (2.08 + 2.08i)T + 47iT^{2} \) |
| 53 | \( 1 + (-10.8 + 2.15i)T + (48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-1.16 - 0.775i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-4.93 - 0.982i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 0.0615i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-1.20 - 2.91i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (5.02 - 12.1i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-9.18 - 9.18i)T + 79iT^{2} \) |
| 83 | \( 1 + (-6.47 - 9.69i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-0.122 + 0.0509i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−10.71665027545226717779745750720, −10.02752239786720928795908071726, −9.576703693783375571342436278952, −8.564479244777458527044846739185, −6.84333519189927763303593594512, −5.47947976779445096250626729721, −5.20133310215317232824969836416, −3.91895043453775995977748869889, −3.07136088951706785833388950954, −1.00959291741860802576632022095,
2.19925319274291615752155883605, 3.15211160263892369163372184583, 4.82985026689070505871989970637, 6.02177890568508671986065154094, 6.71909917316042204306825907431, 7.10101068798732990229689496481, 8.236573099288624664491825291032, 9.522443605418223142798398536442, 10.37662148211908066290024719613, 11.89414043783344281166674881721
