| L(s) = 1 | + (1.04 + 0.948i)2-s + (0.453 + 0.891i)3-s + (0.202 + 1.98i)4-s + (−2.12 + 0.705i)5-s + (−0.368 + 1.36i)6-s + (−1.44 − 1.44i)7-s + (−1.67 + 2.27i)8-s + (−0.587 + 0.809i)9-s + (−2.89 − 1.27i)10-s + (2.44 + 3.37i)11-s + (−1.68 + 1.08i)12-s + (0.588 + 3.71i)13-s + (−0.146 − 2.88i)14-s + (−1.59 − 1.56i)15-s + (−3.91 + 0.804i)16-s + (5.53 + 2.81i)17-s + ⋯ |
| L(s) = 1 | + (0.741 + 0.670i)2-s + (0.262 + 0.514i)3-s + (0.101 + 0.994i)4-s + (−0.948 + 0.315i)5-s + (−0.150 + 0.557i)6-s + (−0.546 − 0.546i)7-s + (−0.592 + 0.805i)8-s + (−0.195 + 0.269i)9-s + (−0.915 − 0.401i)10-s + (0.738 + 1.01i)11-s + (−0.485 + 0.312i)12-s + (0.163 + 1.03i)13-s + (−0.0390 − 0.771i)14-s + (−0.411 − 0.405i)15-s + (−0.979 + 0.201i)16-s + (1.34 + 0.683i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.691933 + 1.51146i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.691933 + 1.51146i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.04 - 0.948i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (2.12 - 0.705i)T \) |
| good | 7 | \( 1 + (1.44 + 1.44i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.44 - 3.37i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.588 - 3.71i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-5.53 - 2.81i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.95 + 6.02i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.04 + 6.56i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.76 - 0.572i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.78 + 2.20i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (5.67 - 0.899i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-9.72 - 7.06i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (0.184 - 0.184i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.87 - 0.953i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (6.30 - 3.21i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.94 - 2.14i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.57 + 3.32i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.89 - 5.67i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (4.27 + 1.38i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (8.01 + 1.27i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (0.543 - 1.67i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-10.4 - 5.30i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (5.72 + 7.87i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (1.84 + 3.61i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.18269750369199030581789244024, −11.36109506289127657877371174282, −10.23905054504807478509342317529, −9.078786890005403034045273930130, −8.098181325267305974957750967406, −7.02128558815102721739647402435, −6.44038308010108901679016586052, −4.57838481036447318751502405322, −4.12942491581620848130259747145, −2.91831591186303369533361654039,
1.04352226494490630739544520440, 3.09533789943498907211317008454, 3.69075705910415961771925594310, 5.38546816698724830972758540073, 6.17445317396714430908131763296, 7.56660074163440170517480645192, 8.567649927413364615821736334647, 9.597010300393099461901267199064, 10.71500118547274611567494716842, 11.87018542989169907637967602356
