L(s) = 1 | + 1.04i·2-s + (6.16 + 6.55i)3-s + 14.8·4-s − 21.6i·5-s + (−6.87 + 6.46i)6-s + 65.3·7-s + 32.4i·8-s + (−4.95 + 80.8i)9-s + 22.6·10-s + 53.0i·11-s + (91.8 + 97.6i)12-s − 148.·13-s + 68.5i·14-s + (141. − 133. i)15-s + 204.·16-s − 477. i·17-s + ⋯ |
L(s) = 1 | + 0.262i·2-s + (0.685 + 0.728i)3-s + 0.931·4-s − 0.865i·5-s + (−0.190 + 0.179i)6-s + 1.33·7-s + 0.506i·8-s + (−0.0611 + 0.998i)9-s + 0.226·10-s + 0.438i·11-s + (0.638 + 0.678i)12-s − 0.877·13-s + 0.349i·14-s + (0.630 − 0.592i)15-s + 0.798·16-s − 1.65i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.339509717\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.339509717\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-6.16 - 6.55i)T \) |
| 59 | \( 1 + 453. iT \) |
good | 2 | \( 1 - 1.04iT - 16T^{2} \) |
| 5 | \( 1 + 21.6iT - 625T^{2} \) |
| 7 | \( 1 - 65.3T + 2.40e3T^{2} \) |
| 11 | \( 1 - 53.0iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 148.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 477. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 527.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 216. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 154. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.32e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.03e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.57e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.24e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.55e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.82e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 - 5.20e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.56e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.35e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.81e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.65e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.09e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 6.02e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.71e3T + 8.85e7T^{2} \) |
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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
−11.79341252802955363091890463807, −11.34693816219604374313652576144, −9.963165230734491284941422409419, −9.110620489067648958667753736183, −7.86037341483765070708225558854, −7.37074238769526762802091647782, −5.24028027641297443575223893727, −4.78610671930238296673522179750, −2.89636409582627169289788212005, −1.56859750196541269062349074673,
1.41572237203655855424236265990, 2.43430798638611023896976086705, 3.61208972629496676573976029183, 5.66370232332631707183934686363, 6.96818207172040734679944451219, 7.59419623013712659394076175075, 8.588649178366865140310099406399, 10.13464579278116561038728041489, 11.04733905514819298755035025616, 11.81920656493862257559480395733