| L(s) = 1 | + (1.39 − 0.200i)2-s + (0.881 − 0.471i)3-s + (1.91 − 0.561i)4-s + (1.14 + 1.39i)5-s + (1.13 − 0.836i)6-s + (−0.882 + 0.589i)7-s + (2.57 − 1.17i)8-s + (0.555 − 0.831i)9-s + (1.88 + 1.72i)10-s + (−0.159 + 0.0483i)11-s + (1.42 − 1.40i)12-s + (−3.93 − 3.22i)13-s + (−1.11 + 1.00i)14-s + (1.66 + 0.691i)15-s + (3.36 − 2.15i)16-s + (−6.65 + 2.75i)17-s + ⋯ |
| L(s) = 1 | + (0.989 − 0.141i)2-s + (0.509 − 0.272i)3-s + (0.959 − 0.280i)4-s + (0.512 + 0.624i)5-s + (0.465 − 0.341i)6-s + (−0.333 + 0.222i)7-s + (0.910 − 0.414i)8-s + (0.185 − 0.277i)9-s + (0.595 + 0.545i)10-s + (−0.0480 + 0.0145i)11-s + (0.412 − 0.404i)12-s + (−1.09 − 0.895i)13-s + (−0.298 + 0.267i)14-s + (0.430 + 0.178i)15-s + (0.842 − 0.539i)16-s + (−1.61 + 0.668i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.85694 - 0.330473i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.85694 - 0.330473i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.200i)T \) |
| 3 | \( 1 + (-0.881 + 0.471i)T \) |
| good | 5 | \( 1 + (-1.14 - 1.39i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (0.882 - 0.589i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.159 - 0.0483i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (3.93 + 3.22i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (6.65 - 2.75i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.712 - 7.23i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-5.33 + 1.06i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.91 + 6.32i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (3.85 + 3.85i)T + 31iT^{2} \) |
| 37 | \( 1 + (7.55 + 0.743i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.713 - 3.58i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-1.35 - 0.725i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (0.849 + 2.05i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (2.11 + 6.97i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-0.186 + 0.152i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-6.44 - 12.0i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-4.78 - 8.94i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-3.26 - 4.88i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (10.3 + 6.92i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (6.08 - 14.6i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.98 + 0.293i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.54 - 1.10i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (2.74 + 2.74i)T + 97iT^{2} \) |
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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
−11.42350425409598850442582122584, −10.38418715872737413719766429726, −9.820583148909264764530174043124, −8.382741191680050656472930556457, −7.29126814546860676568892787038, −6.43587052030481235195439025752, −5.56378394303609808011028736057, −4.19770082031512060557785544682, −2.94022988567139220082890597208, −2.10104871591133287362006138068,
2.07064025073205068287236101661, 3.22230548668113310712433603144, 4.76629591246264257235912903657, 5.02411739804789412397758148005, 6.82915271058160767443484512281, 7.13939652004483239828586120895, 8.900913350220074907071757226265, 9.279126099394110758881731425170, 10.65761560719971130826995988766, 11.43568986128763602003493898373
