L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−7.61 − 13.1i)5-s − 7.99·8-s + (15.2 − 26.3i)10-s + (1 − 1.73i)11-s + 30.4·13-s + (−8 − 13.8i)16-s + (22.8 − 39.5i)17-s + (76.1 + 131. i)19-s + 60.9·20-s + 3.99·22-s + (15 + 25.9i)23-s + (−53.5 + 92.6i)25-s + (30.4 + 52.7i)26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.681 − 1.17i)5-s − 0.353·8-s + (0.481 − 0.834i)10-s + (0.0274 − 0.0474i)11-s + 0.649·13-s + (−0.125 − 0.216i)16-s + (0.325 − 0.564i)17-s + (0.919 + 1.59i)19-s + 0.681·20-s + 0.0387·22-s + (0.135 + 0.235i)23-s + (−0.427 + 0.741i)25-s + (0.229 + 0.397i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8944145816\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8944145816\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (7.61 + 13.1i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 30.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-22.8 + 39.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-76.1 - 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-15 - 25.9i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 212T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-106. + 184. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (123 + 213. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 284T + 7.95e4T^{2} \) |
| 47 | \( 1 + (30.4 + 52.7i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-274 + 474. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-335. + 580. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (258. + 448. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (326 - 564. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 770T + 3.57e5T^{2} \) |
| 73 | \( 1 + (487. - 844. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (236 + 408. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 182.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (357. + 619. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 304.T + 9.12e5T^{2} \) |
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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
−9.300923696911841083578580121172, −8.399531734441826551711448965599, −7.902268950217374564635547478023, −7.00729932592198855651276505919, −5.73963712269454542812193534845, −5.21584634831465477775720968772, −4.08872921556467143489206129006, −3.42835621546613244156668145739, −1.53336759996161026529063834937, −0.21565718342853067837601354574,
1.36432109454915316902918941861, 2.87475266732573654092380543068, 3.39341565055083661245140394450, 4.47634909195130006488763853507, 5.56537778095976981885457314198, 6.68407715637186431101662847553, 7.27842424467783372189959939292, 8.399683462878419090482453879421, 9.277353236185372818619942280541, 10.40009087406352208745289153961