| L(s) = 1 | + (1.15 + 0.669i)5-s + (0.328 + 0.569i)7-s + (7.33 − 4.23i)11-s + (0.615 − 1.06i)13-s − 5.78i·17-s − 22.7·19-s + (−17.4 − 10.0i)23-s + (−11.6 − 20.0i)25-s + (−35.5 + 20.5i)29-s + (7.57 − 13.1i)31-s + 0.880i·35-s + 51.2·37-s + (−17.7 − 10.2i)41-s + (−3.14 − 5.44i)43-s + (−31.9 + 18.4i)47-s + ⋯ |
| L(s) = 1 | + (0.231 + 0.133i)5-s + (0.0469 + 0.0813i)7-s + (0.667 − 0.385i)11-s + (0.0473 − 0.0820i)13-s − 0.340i·17-s − 1.19·19-s + (−0.759 − 0.438i)23-s + (−0.464 − 0.803i)25-s + (−1.22 + 0.707i)29-s + (0.244 − 0.423i)31-s + 0.0251i·35-s + 1.38·37-s + (−0.433 − 0.250i)41-s + (−0.0731 − 0.126i)43-s + (−0.679 + 0.392i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8775375234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8775375234\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.15 - 0.669i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-0.328 - 0.569i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.33 + 4.23i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-0.615 + 1.06i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 5.78iT - 289T^{2} \) |
| 19 | \( 1 + 22.7T + 361T^{2} \) |
| 23 | \( 1 + (17.4 + 10.0i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (35.5 - 20.5i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-7.57 + 13.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 51.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (17.7 + 10.2i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (3.14 + 5.44i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (31.9 - 18.4i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 85.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (51.2 + 29.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-38.1 - 66.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-48.2 + 83.5i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 31.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 24.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + (67.4 + 116. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (98.6 - 56.9i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (68.8 + 119. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−8.837159582231721964034852162097, −8.143881230038484656014383825957, −7.23903318354130576533743912370, −6.28256316712497027201633856016, −5.83319607072369763172000038760, −4.58494248978596533476636689175, −3.86323775325497069497957960744, −2.69199394026881688037533502777, −1.69239243662994862507572769025, −0.22099152820205443602640796525,
1.41191034188247970076332982206, 2.30650466344235115547820946111, 3.71626097105741124969049464835, 4.32425717058922371558909731794, 5.45333059821856596981290873422, 6.21891392062135928857764163549, 6.98732558351878574230551850675, 7.927218356887296925436313170710, 8.604788540417137898425846074914, 9.600213186546450604580282076305
