| L(s) = 1 | + (1.06 + 0.924i)2-s + (−1.64 + 0.544i)3-s + (0.288 + 1.97i)4-s + (−0.372 + 1.22i)5-s + (−2.26 − 0.937i)6-s + (0.186 + 0.939i)7-s + (−1.52 + 2.38i)8-s + (2.40 − 1.79i)9-s + (−1.53 + 0.969i)10-s + (−2.85 + 0.281i)11-s + (−1.55 − 3.09i)12-s + (0.188 + 0.619i)13-s + (−0.669 + 1.17i)14-s + (−0.0566 − 2.22i)15-s + (−3.83 + 1.14i)16-s + (−2.16 − 0.897i)17-s + ⋯ |
| L(s) = 1 | + (0.756 + 0.654i)2-s + (−0.949 + 0.314i)3-s + (0.144 + 0.989i)4-s + (−0.166 + 0.549i)5-s + (−0.923 − 0.382i)6-s + (0.0706 + 0.355i)7-s + (−0.537 + 0.843i)8-s + (0.802 − 0.597i)9-s + (−0.485 + 0.306i)10-s + (−0.861 + 0.0848i)11-s + (−0.448 − 0.893i)12-s + (0.0521 + 0.171i)13-s + (−0.178 + 0.314i)14-s + (−0.0146 − 0.573i)15-s + (−0.958 + 0.285i)16-s + (−0.525 − 0.217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.116631 + 1.14308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.116631 + 1.14308i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.06 - 0.924i)T \) |
| 3 | \( 1 + (1.64 - 0.544i)T \) |
| good | 5 | \( 1 + (0.372 - 1.22i)T + (-4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (-0.186 - 0.939i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (2.85 - 0.281i)T + (10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-0.188 - 0.619i)T + (-10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (2.16 + 0.897i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (1.99 - 3.73i)T + (-10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (3.32 + 4.97i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-2.82 - 0.277i)T + (28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (-4.18 - 4.18i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.404 - 0.756i)T + (-20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (-1.92 - 2.88i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-6.89 - 5.66i)T + (8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (1.22 + 0.508i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-7.63 + 0.752i)T + (51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (1.97 - 6.49i)T + (-49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (5.87 - 4.82i)T + (11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (2.69 + 3.27i)T + (-13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (-1.51 - 7.63i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-15.2 - 3.03i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (1.60 + 3.87i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-6.83 + 12.7i)T + (-46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-0.477 - 0.318i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-2.88 + 2.88i)T - 97iT^{2} \) |
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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
−11.95106424838075210501794579889, −10.93730188810791800949092485992, −10.26828900565944060173461242163, −8.842920430779626802897172154349, −7.77490165999209352145303324786, −6.73256537730652074773155138482, −6.03677695272372207585985492929, −5.00401643625444714095565658291, −4.11492185670636508316984688894, −2.67263648429097772545908191650,
0.66370648075447537673276171655, 2.31861356343032496579651672199, 4.09566196300574177872621107668, 4.93303277441524094749831738931, 5.82351878194644158805129478665, 6.83677380338021240844075879245, 7.986460478167861511171703934816, 9.369535299266161018902755136072, 10.50915560923533638674961552599, 10.93763696892520701170527435668
