| L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.587 + 0.809i)3-s + (0.997 + 1.73i)4-s + (0.0244 − 0.0751i)5-s + (−1.29 + 0.574i)6-s + (−0.500 + 0.363i)7-s + (−0.00503 + 2.82i)8-s + (−0.309 − 0.951i)9-s + (0.0830 − 0.0747i)10-s + (−0.0857 − 3.31i)11-s + (−1.98 − 0.211i)12-s + (3.41 − 1.10i)13-s + (−0.870 + 0.0909i)14-s + (0.0464 + 0.0639i)15-s + (−2.00 + 3.45i)16-s + (−2.15 − 0.700i)17-s + ⋯ |
| L(s) = 1 | + (0.865 + 0.500i)2-s + (−0.339 + 0.467i)3-s + (0.498 + 0.866i)4-s + (0.0109 − 0.0336i)5-s + (−0.527 + 0.234i)6-s + (−0.189 + 0.137i)7-s + (−0.00178 + 0.999i)8-s + (−0.103 − 0.317i)9-s + (0.0262 − 0.0236i)10-s + (−0.0258 − 0.999i)11-s + (−0.574 − 0.0610i)12-s + (0.946 − 0.307i)13-s + (−0.232 + 0.0243i)14-s + (0.0119 + 0.0165i)15-s + (−0.502 + 0.864i)16-s + (−0.522 − 0.169i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25262 + 0.855655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25262 + 0.855655i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 + (0.0857 + 3.31i)T \) |
| good | 5 | \( 1 + (-0.0244 + 0.0751i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (0.500 - 0.363i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-3.41 + 1.10i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.15 + 0.700i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.03 + 0.750i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 5.24iT - 23T^{2} \) |
| 29 | \( 1 + (4.38 + 6.03i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.81 - 0.914i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.85 + 1.34i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (7.06 - 9.72i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 8.23T + 43T^{2} \) |
| 47 | \( 1 + (4.07 - 5.61i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.742 + 2.28i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.18 - 1.63i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (9.53 + 3.09i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 10.2iT - 67T^{2} \) |
| 71 | \( 1 + (-9.14 - 2.97i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (5.36 + 7.38i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.38 - 7.33i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.80 + 5.54i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 3.55T + 89T^{2} \) |
| 97 | \( 1 + (-4.20 - 12.9i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.43287848214164260539893259550, −12.71347140960432639204168865889, −11.37832174332184010391656004697, −10.82894863884719053119140322038, −9.108544640186681032121462704351, −8.096521772111759266719711770511, −6.53979925389775708924527348151, −5.72825963498425668151857811604, −4.40697011332759607975299775209, −3.09515354195315454160384599414,
1.84401657708715993629582173946, 3.72139725441163026299429814360, 5.11079951871015561712695668114, 6.36873533082844174734833030487, 7.29244203509245886537979906491, 9.065932636505861068298075407638, 10.37676138200668492211696206977, 11.20182002764457450126737515015, 12.23988754748889424713465585568, 13.00222186096419018689544644863
