| L(s) = 1 | + (−1.41 − 0.0418i)2-s + (−0.0592 − 0.297i)3-s + (1.99 + 0.118i)4-s + (−0.919 + 1.37i)5-s + (0.0712 + 0.423i)6-s + (1.30 + 1.94i)7-s + (−2.81 − 0.250i)8-s + (2.68 − 1.11i)9-s + (1.35 − 1.90i)10-s + (0.187 + 0.0373i)11-s + (−0.0830 − 0.601i)12-s + (3.45 + 3.45i)13-s + (−1.75 − 2.80i)14-s + (0.464 + 0.192i)15-s + (3.97 + 0.472i)16-s + (3.91 + 1.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0296i)2-s + (−0.0341 − 0.171i)3-s + (0.998 + 0.0591i)4-s + (−0.411 + 0.615i)5-s + (0.0290 + 0.172i)6-s + (0.492 + 0.736i)7-s + (−0.996 − 0.0887i)8-s + (0.895 − 0.370i)9-s + (0.429 − 0.602i)10-s + (0.0565 + 0.0112i)11-s + (−0.0239 − 0.173i)12-s + (0.957 + 0.957i)13-s + (−0.470 − 0.750i)14-s + (0.119 + 0.0496i)15-s + (0.992 + 0.118i)16-s + (0.949 + 0.314i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.725781 + 0.191549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.725781 + 0.191549i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0418i)T \) |
| 17 | \( 1 + (-3.91 - 1.29i)T \) |
| good | 3 | \( 1 + (0.0592 + 0.297i)T + (-2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (0.919 - 1.37i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 1.94i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.187 - 0.0373i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-3.45 - 3.45i)T + 13iT^{2} \) |
| 19 | \( 1 + (2.93 + 1.21i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (2.37 + 0.473i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (4.22 + 2.82i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (1.98 + 9.97i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (5.09 - 1.01i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (1.88 - 1.25i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (5.01 - 2.07i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-3.85 - 3.85i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.05 + 7.38i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.53 - 6.12i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-8.83 + 5.90i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + 3.36iT - 67T^{2} \) |
| 71 | \( 1 + (9.26 - 1.84i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (2.65 + 1.77i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.441 - 2.21i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-5.04 + 12.1i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (3.00 + 3.00i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.01 - 6.01i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−13.09816535760394812753409273811, −11.86505593096136281569256622570, −11.31567412701504809670560435104, −10.15685651189254839588661568493, −9.100005688493532353603250013505, −8.052416689930029223697642405249, −7.01874355021494324056084494147, −5.98291263044907572877291108875, −3.77972446196444411179786605549, −1.87378356416264616866752713960,
1.30038044188279376961800053056, 3.74431367182527081200224350316, 5.36028047875034617575031051988, 7.03243399907613829381350363416, 7.968592959444308752325684750816, 8.786627690688922819562889036007, 10.29027822786085382098768071800, 10.64956756523356048233419439499, 11.99940401869240734461368461349, 12.90699297330506527405249076369
