| L(s) = 1 | + (1.08 − 0.787i)2-s + (−1.91 + 5.90i)4-s + (3.83 + 10.5i)5-s − 12.2·7-s + (5.88 + 18.0i)8-s + (12.4 + 8.36i)10-s + (−2.00 + 1.45i)11-s + (−69.2 − 50.3i)13-s + (−13.2 + 9.61i)14-s + (−19.5 − 14.1i)16-s + (1.71 + 5.27i)17-s + (−7.45 − 22.9i)19-s + (−69.3 + 2.47i)20-s + (−1.02 + 3.16i)22-s + (−83.6 + 60.7i)23-s + ⋯ |
| L(s) = 1 | + (0.383 − 0.278i)2-s + (−0.239 + 0.737i)4-s + (0.342 + 0.939i)5-s − 0.659·7-s + (0.259 + 0.799i)8-s + (0.392 + 0.264i)10-s + (−0.0550 + 0.0399i)11-s + (−1.47 − 1.07i)13-s + (−0.252 + 0.183i)14-s + (−0.305 − 0.221i)16-s + (0.0244 + 0.0752i)17-s + (−0.0900 − 0.277i)19-s + (−0.775 + 0.0276i)20-s + (−0.00995 + 0.0306i)22-s + (−0.758 + 0.550i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.197575 + 0.902010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.197575 + 0.902010i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-3.83 - 10.5i)T \) |
| good | 2 | \( 1 + (-1.08 + 0.787i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 + 12.2T + 343T^{2} \) |
| 11 | \( 1 + (2.00 - 1.45i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (69.2 + 50.3i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-1.71 - 5.27i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (7.45 + 22.9i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (83.6 - 60.7i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-35.3 + 108. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-47.5 - 146. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-277. - 201. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-28.0 - 20.3i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (104. - 321. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (199. - 614. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-536. - 389. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-289. + 210. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (164. + 505. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-196. + 603. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (779. - 565. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-175. + 540. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-98.8 - 304. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (1.07e3 - 784. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-123. + 379. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.31112640909769852428647175044, −11.39779073341380704069077145458, −10.23619485917624639043575931379, −9.560640237658548256532870299467, −8.052936488561918697346323979417, −7.26820731700159343726627246837, −6.03726179558579119981536277698, −4.71389531183316385930775208756, −3.26351969175003447118674267473, −2.52916221994782244012203686361,
0.31520791500618934379341587588, 2.04493539823985232511454061049, 4.15775197847788724245535886672, 5.04296155640296357820258891408, 6.07857582088950379746589517486, 7.06531912319862631734159956173, 8.556875563182746467097226676362, 9.756030821494075212744111794787, 9.886531872123558892147995532870, 11.55868013488624062382000616906
