| L(s) = 1 | + (−1.41 − 0.0249i)2-s + (−1.51 + 0.301i)3-s + (1.99 + 0.0705i)4-s + (1.90 + 1.27i)5-s + (2.14 − 0.387i)6-s + (2.53 + 3.79i)7-s + (−2.82 − 0.149i)8-s + (−0.571 + 0.236i)9-s + (−2.66 − 1.85i)10-s + (0.737 − 3.70i)11-s + (−3.04 + 0.494i)12-s + (1.06 − 1.06i)13-s + (−3.48 − 5.42i)14-s + (−3.27 − 1.35i)15-s + (3.99 + 0.282i)16-s + (−3.21 + 2.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0176i)2-s + (−0.873 + 0.173i)3-s + (0.999 + 0.0352i)4-s + (0.853 + 0.570i)5-s + (0.876 − 0.158i)6-s + (0.957 + 1.43i)7-s + (−0.998 − 0.0529i)8-s + (−0.190 + 0.0789i)9-s + (−0.843 − 0.585i)10-s + (0.222 − 1.11i)11-s + (−0.879 + 0.142i)12-s + (0.296 − 0.296i)13-s + (−0.932 − 1.44i)14-s + (−0.845 − 0.350i)15-s + (0.997 + 0.0705i)16-s + (−0.779 + 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.520859 + 0.214305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.520859 + 0.214305i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0249i)T \) |
| 17 | \( 1 + (3.21 - 2.58i)T \) |
| good | 3 | \( 1 + (1.51 - 0.301i)T + (2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (-1.90 - 1.27i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-2.53 - 3.79i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.737 + 3.70i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-1.06 + 1.06i)T - 13iT^{2} \) |
| 19 | \( 1 + (-0.308 + 0.745i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (1.55 + 0.310i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.05 + 1.58i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (0.416 + 2.09i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (1.44 + 7.27i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-6.06 + 4.05i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.146 - 0.352i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (0.465 + 0.465i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.45 - 1.01i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (5.44 - 2.25i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (3.92 + 5.87i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + 16.2T + 67T^{2} \) |
| 71 | \( 1 + (-7.36 + 1.46i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-5.84 - 3.90i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (2.33 - 11.7i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-6.25 - 2.58i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-5.52 - 5.52i)T + 89iT^{2} \) |
| 97 | \( 1 + (-3.39 + 5.07i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−15.18535837902726527106287051095, −14.05415381749113296976681783476, −12.22268087736603725050397734256, −11.20085187346036909477326592863, −10.69782353451495956231459469501, −9.133117362743938011073849057965, −8.226861910876183682532151352848, −6.19748952378691146245603493026, −5.66651539471928831147833361887, −2.36832519406400931961134308438,
1.44375205495162125440148858811, 4.83303884933705100038503831165, 6.41856843589276254702038110342, 7.51385808623818504879125737049, 9.016834867824573991682563177154, 10.18829701484952609777604880344, 11.12953210913370861943188600429, 12.08615365355207320183099039624, 13.53489064429796617671822462338, 14.71420106783982098646816801307
