| 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
−11.63915348166915758201766144196, −11.33363486294219190844660849090, −9.568748336942547204334674761580, −9.185122119245939994629363504052, −7.990232922773917656635717414964, −7.12052073379266067108719941558, −5.06908084937627607125728997302, −4.67847461169587960116385172179, −3.03899555052037373321154752956, −1.25904119509578304936101088978,
0.968450782761713104322542683334, 3.81833018582620290773224957320, 4.83900154949968153258845998820, 6.02030548072244140302011149901, 7.17324255340563920612500350392, 7.75184453123539411904427553294, 8.576766017712978307214452450645, 10.28570319129009901694534118531, 10.55399451779023488554717291852, 11.67269807085937534601173982624
