| L(s) = 1 | + (0.0546 − 0.729i)2-s + (−1.13 + 0.770i)3-s + (1.44 + 0.218i)4-s + (0.323 − 1.41i)5-s + (0.500 + 0.866i)6-s + (1.33 − 2.28i)7-s + (0.564 − 2.47i)8-s + (−0.412 + 1.05i)9-s + (−1.01 − 0.313i)10-s + (−0.615 + 1.56i)11-s + (−1.80 + 0.868i)12-s + (1.37 + 1.27i)13-s + (−1.59 − 1.09i)14-s + (0.727 + 1.85i)15-s + (1.02 + 0.316i)16-s + (−1.43 − 6.28i)17-s + ⋯ |
| L(s) = 1 | + (0.0386 − 0.516i)2-s + (−0.652 + 0.444i)3-s + (0.723 + 0.109i)4-s + (0.144 − 0.634i)5-s + (0.204 + 0.353i)6-s + (0.504 − 0.863i)7-s + (0.199 − 0.874i)8-s + (−0.137 + 0.350i)9-s + (−0.321 − 0.0992i)10-s + (−0.185 + 0.472i)11-s + (−0.520 + 0.250i)12-s + (0.382 + 0.354i)13-s + (−0.426 − 0.293i)14-s + (0.187 + 0.478i)15-s + (0.256 + 0.0790i)16-s + (−0.347 − 1.52i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24085 - 0.609499i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24085 - 0.609499i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.33 + 2.28i)T \) |
| 43 | \( 1 + (-3.55 - 5.51i)T \) |
| good | 2 | \( 1 + (-0.0546 + 0.729i)T + (-1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (1.13 - 0.770i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.323 + 1.41i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (0.615 - 1.56i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.37 - 1.27i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.43 + 6.28i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (-4.72 + 5.92i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (-2.35 - 2.95i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (0.326 - 4.35i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (0.182 - 2.43i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 + (7.10 - 3.42i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-2.74 - 6.99i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (4.03 + 1.24i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (6.82 - 6.33i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.462 - 6.16i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (2.37 + 6.04i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (3.67 + 9.37i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (3.03 - 13.3i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (-8.23 - 14.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.190 + 2.54i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (3.47 + 1.67i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (7.72 + 9.68i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.26101496095320771200215461636, −11.05819739513821479107689957212, −9.969868619394182526666479164567, −9.021367886457225185884842821535, −7.50284828035805799256204551419, −6.87508410724120943575012466816, −5.20546201317813554479659213959, −4.62647934046458681065010460286, −2.99476761921623460215511085216, −1.27345200240316957567852460646,
1.81079514804635991328348881259, 3.31086315924631397831076586612, 5.44152789783050003146942163019, 5.99664973183250435931710896827, 6.73994681967026897496393211270, 7.911350312263125247553906371431, 8.722208844963500754262844586109, 10.38268800115509194672853004965, 10.97322744694734385377369736252, 11.91175733896107674564858914106
