| L(s) = 1 | + (−0.557 + 1.29i)2-s + (−0.987 − 0.156i)3-s + (−1.37 − 1.44i)4-s + (−1.94 + 1.09i)5-s + (0.753 − 1.19i)6-s + (2.52 − 2.52i)7-s + (2.65 − 0.984i)8-s + (0.951 + 0.309i)9-s + (−0.342 − 3.14i)10-s + (1.68 − 0.548i)11-s + (1.13 + 1.64i)12-s + (−1.62 − 3.19i)13-s + (1.87 + 4.68i)14-s + (2.09 − 0.780i)15-s + (−0.197 + 3.99i)16-s + (0.815 + 5.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.394 + 0.919i)2-s + (−0.570 − 0.0903i)3-s + (−0.689 − 0.724i)4-s + (−0.871 + 0.491i)5-s + (0.307 − 0.488i)6-s + (0.954 − 0.954i)7-s + (0.937 − 0.348i)8-s + (0.317 + 0.103i)9-s + (−0.108 − 0.994i)10-s + (0.509 − 0.165i)11-s + (0.327 + 0.475i)12-s + (−0.451 − 0.885i)13-s + (0.500 + 1.25i)14-s + (0.541 − 0.201i)15-s + (−0.0493 + 0.998i)16-s + (0.197 + 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.792387 + 0.175928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.792387 + 0.175928i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.557 - 1.29i)T \) |
| 3 | \( 1 + (0.987 + 0.156i)T \) |
| 5 | \( 1 + (1.94 - 1.09i)T \) |
| good | 7 | \( 1 + (-2.52 + 2.52i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.68 + 0.548i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.62 + 3.19i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.815 - 5.14i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-6.47 - 4.70i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.63 + 7.12i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-2.24 - 3.09i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.52 + 3.47i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 2.07i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-1.90 + 5.86i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (6.90 + 6.90i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.113 + 0.716i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (0.150 - 0.949i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-0.117 + 0.360i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.14 - 9.67i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (13.5 - 2.14i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (0.772 + 1.06i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.731 - 0.372i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-5.19 + 3.77i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (0.109 + 0.694i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (4.92 - 1.60i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.43 - 1.33i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.67269807085937534601173982624, −10.55399451779023488554717291852, −10.28570319129009901694534118531, −8.576766017712978307214452450645, −7.75184453123539411904427553294, −7.17324255340563920612500350392, −6.02030548072244140302011149901, −4.83900154949968153258845998820, −3.81833018582620290773224957320, −0.968450782761713104322542683334,
1.25904119509578304936101088978, 3.03899555052037373321154752956, 4.67847461169587960116385172179, 5.06908084937627607125728997302, 7.12052073379266067108719941558, 7.990232922773917656635717414964, 9.185122119245939994629363504052, 9.568748336942547204334674761580, 11.33363486294219190844660849090, 11.63915348166915758201766144196
