| L(s) = 1 | + (0.415 + 0.909i)2-s + (0.461 + 0.296i)3-s + (−0.654 + 0.755i)4-s + (−0.142 − 0.989i)5-s + (−0.0780 + 0.543i)6-s + (−1.43 − 3.14i)7-s + (−0.959 − 0.281i)8-s + (−1.12 − 2.45i)9-s + (0.841 − 0.540i)10-s + (−0.0589 − 0.410i)11-s + (−0.526 + 0.154i)12-s + (−4.79 + 1.40i)13-s + (2.26 − 2.61i)14-s + (0.227 − 0.499i)15-s + (−0.142 − 0.989i)16-s + (−2.03 − 2.34i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (0.266 + 0.171i)3-s + (−0.327 + 0.377i)4-s + (−0.0636 − 0.442i)5-s + (−0.0318 + 0.221i)6-s + (−0.542 − 1.18i)7-s + (−0.339 − 0.0996i)8-s + (−0.373 − 0.818i)9-s + (0.266 − 0.170i)10-s + (−0.0177 − 0.123i)11-s + (−0.151 + 0.0446i)12-s + (−1.33 + 0.390i)13-s + (0.604 − 0.697i)14-s + (0.0588 − 0.128i)15-s + (−0.0355 − 0.247i)16-s + (−0.493 − 0.569i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0942 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0942 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.689114 - 0.626947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.689114 - 0.626947i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (-6.39 - 5.10i)T \) |
| good | 3 | \( 1 + (-0.461 - 0.296i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (1.43 + 3.14i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.0589 + 0.410i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (4.79 - 1.40i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (2.03 + 2.34i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (0.213 - 0.467i)T + (-12.4 - 14.3i)T^{2} \) |
| 23 | \( 1 + (5.35 + 3.44i)T + (9.55 + 20.9i)T^{2} \) |
| 29 | \( 1 + 1.89T + 29T^{2} \) |
| 31 | \( 1 + (-2.07 - 0.608i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + (-5.37 - 6.20i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.03 + 1.19i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-8.03 - 5.16i)T + (19.5 + 42.7i)T^{2} \) |
| 53 | \( 1 + (-1.66 + 1.92i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.52 - 1.03i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.88 + 13.0i)T + (-58.5 - 17.1i)T^{2} \) |
| 71 | \( 1 + (6.26 - 7.22i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.46 + 10.1i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (7.75 - 2.27i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.24 + 8.65i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-2.83 + 1.82i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−9.913820803058803456917481571701, −9.533016627253465589401851644434, −8.492852105015662349305607155111, −7.55824561623072048682145037813, −6.79185476828967854528503180924, −5.92541591884090694679970558074, −4.57381021431349689108068725638, −4.01073297493635909711762424988, −2.72473907354064481892290762798, −0.40677778415458192287551617181,
2.26737282708400433185949519202, 2.64944807012938409232801455818, 4.04406894734883399011368583770, 5.32820046799941209242507121550, 5.98125104713609751719032933001, 7.27938869500973096163467882849, 8.182545448783979204909702645710, 9.150223361811273150434044914741, 9.916771563387496104668512217282, 10.72171332119596299523273568093
