L(s) = 1 | + (−3.16 + 1.82i)2-s + (2.67 − 4.63i)4-s + (8.69 + 15.0i)5-s − 9.68i·8-s + (−55.0 − 31.7i)10-s + (28.2 + 16.2i)11-s − 42.4i·13-s + (39.0 + 67.7i)16-s + (42.8 − 74.1i)17-s + (59.9 − 34.5i)19-s + 93.0·20-s − 118.·22-s + (144. − 83.5i)23-s + (−88.8 + 153. i)25-s + (77.5 + 134. i)26-s + ⋯ |
L(s) = 1 | + (−1.11 + 0.645i)2-s + (0.334 − 0.578i)4-s + (0.778 + 1.34i)5-s − 0.428i·8-s + (−1.74 − 1.00i)10-s + (0.773 + 0.446i)11-s − 0.906i·13-s + (0.610 + 1.05i)16-s + (0.610 − 1.05i)17-s + (0.723 − 0.417i)19-s + 1.04·20-s − 1.15·22-s + (1.31 − 0.757i)23-s + (−0.710 + 1.23i)25-s + (0.585 + 1.01i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.206820399\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.206820399\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (3.16 - 1.82i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-8.69 - 15.0i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-28.2 - 16.2i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 42.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-42.8 + 74.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-59.9 + 34.5i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-144. + 83.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 254. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (281. + 162. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (172. + 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 182.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 140.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (21.5 + 37.2i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (167. + 96.5i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-55.6 + 96.3i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-78.4 + 45.2i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-524. + 908. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 464. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (158. + 91.3i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-119. - 206. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 375.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-719. - 1.24e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 638. iT - 9.12e5T^{2} \) |
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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
−10.53571825602875351658619945612, −9.524103614442068039797011827767, −9.310134399908928697419258092541, −7.78855639962708707460826995344, −7.19454481682458658543742141864, −6.45298963968516840878211218039, −5.43502675692186260488876167275, −3.61613509790943999840056563465, −2.40369880524463715791654012290, −0.67206965453904662960392939330,
1.27565160910564027245926963118, 1.52765128046938749763621886741, 3.41812570374616467711014901152, 4.98648131366442863644946067500, 5.74788672399074873893193619225, 7.17186039025879220953218706844, 8.481824949483306326916359454344, 9.009159704395064007430481604910, 9.493213964025985155488429374563, 10.47614039501964695618834511441