| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 42.3i·5-s + 22.6i·8-s − 119.·10-s + 183. i·11-s − 98.5·13-s + 64.0·16-s − 532. i·17-s + 296.·19-s + 339. i·20-s + 518.·22-s + 878. i·23-s − 1.17e3·25-s + 278. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.69i·5-s + 0.353i·8-s − 1.19·10-s + 1.51i·11-s − 0.583·13-s + 0.250·16-s − 1.84i·17-s + 0.822·19-s + 0.847i·20-s + 1.07·22-s + 1.65i·23-s − 1.87·25-s + 0.412i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.586838635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586838635\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 42.3iT - 625T^{2} \) |
| 11 | \( 1 - 183. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 98.5T + 2.85e4T^{2} \) |
| 17 | \( 1 + 532. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 296.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 878. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 849. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 411.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.45e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.52e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.08e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 944. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 206. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.53e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 179.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 607.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.05e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.42e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.33e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.34e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.07e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.48e3T + 8.85e7T^{2} \) |
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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
−9.410802865716000198909388739816, −9.127565616977570451955191897476, −7.76068564272990111530263035810, −7.27472195812652093253916081918, −5.50940502568059019342574891213, −4.91423226539186238124131617076, −4.30813360523238413467661362932, −2.89384448165832735789325109979, −1.66544488010159050110703426103, −0.847767434395472870893170096668,
0.46039362248351792785518554112, 2.32407496628142489938984250716, 3.31689802385842720283045693230, 4.15800358651157319646338681511, 5.74948201240485124886551653519, 6.17204409735632499109109926837, 7.00822705678459769638736252313, 7.87423602303194227060049969690, 8.562202743790270081519695637638, 9.695625091771553659972678647993
