| L(s) = 1 | + (−1.41 − 0.0597i)2-s + (0.192 − 1.95i)3-s + (1.99 + 0.168i)4-s + (−1.99 + 1.06i)5-s + (−0.388 + 2.75i)6-s + (0.195 + 0.980i)7-s + (−2.80 − 0.357i)8-s + (−0.846 − 0.168i)9-s + (2.87 − 1.38i)10-s + (−0.00199 + 0.00242i)11-s + (0.713 − 3.86i)12-s + (0.274 − 0.513i)13-s + (−0.217 − 1.39i)14-s + (1.69 + 4.09i)15-s + (3.94 + 0.672i)16-s + (0.0360 − 0.0870i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0422i)2-s + (0.111 − 1.12i)3-s + (0.996 + 0.0843i)4-s + (−0.890 + 0.475i)5-s + (−0.158 + 1.12i)6-s + (0.0737 + 0.370i)7-s + (−0.991 − 0.126i)8-s + (−0.282 − 0.0561i)9-s + (0.909 − 0.437i)10-s + (−0.000600 + 0.000731i)11-s + (0.206 − 1.11i)12-s + (0.0761 − 0.142i)13-s + (−0.0580 − 0.373i)14-s + (0.438 + 1.05i)15-s + (0.985 + 0.168i)16-s + (0.00874 − 0.0211i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0360382 + 0.0714960i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0360382 + 0.0714960i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0597i)T \) |
| 7 | \( 1 + (-0.195 - 0.980i)T \) |
| good | 3 | \( 1 + (-0.192 + 1.95i)T + (-2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (1.99 - 1.06i)T + (2.77 - 4.15i)T^{2} \) |
| 11 | \( 1 + (0.00199 - 0.00242i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.274 + 0.513i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.0360 + 0.0870i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (2.16 + 7.12i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (5.58 - 3.73i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (7.85 - 6.44i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (1.17 - 1.17i)T - 31iT^{2} \) |
| 37 | \( 1 + (10.1 + 3.09i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-3.00 - 4.50i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.206 - 2.10i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (2.90 + 1.20i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.45 - 2.01i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-4.09 - 7.66i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (2.10 + 0.207i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-3.49 - 0.344i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (0.215 - 0.0428i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (2.34 - 11.8i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-10.4 + 4.34i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (0.935 - 0.283i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (9.02 + 6.03i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-1.07 + 1.07i)T - 97iT^{2} \) |
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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
−10.50872762498155717704735480124, −9.363734573080680412332260617388, −8.603318446626248032931202647902, −7.76378727136634195050224476049, −7.21526413426131026152672576093, −6.63246093872219001900166847572, −5.47798189473925442625507850063, −3.76460795663496968793251086767, −2.62211221982379576013990487743, −1.56755403252513049107785626367,
0.05080394841645116033125487940, 1.89846855405755255782279926126, 3.68972028840632481470696583407, 4.08535222877569059515986652578, 5.42211004113436410533286358803, 6.50725052257592799714303999723, 7.69639529285770141079340570639, 8.198046142231552960079938531204, 9.022878758957953272733056569203, 9.918965070375682723603407385236
