| L(s) = 1 | + (0.348 − 2.31i)2-s + (−1.20 − 1.24i)3-s + (−3.32 − 1.02i)4-s + (−2.39 + 1.63i)5-s + (−3.29 + 2.35i)6-s + (1.38 − 2.25i)7-s + (−1.49 + 3.10i)8-s + (−0.0903 + 2.99i)9-s + (2.94 + 6.12i)10-s + (−0.509 + 3.38i)11-s + (2.73 + 5.36i)12-s + (−4.98 + 1.95i)13-s + (−4.73 − 3.99i)14-s + (4.92 + 1.00i)15-s + (0.930 + 0.634i)16-s + (−0.598 + 2.62i)17-s + ⋯ |
| L(s) = 1 | + (0.246 − 1.63i)2-s + (−0.696 − 0.717i)3-s + (−1.66 − 0.512i)4-s + (−1.07 + 0.731i)5-s + (−1.34 + 0.962i)6-s + (0.523 − 0.851i)7-s + (−0.529 + 1.09i)8-s + (−0.0301 + 0.999i)9-s + (0.932 + 1.93i)10-s + (−0.153 + 1.01i)11-s + (0.788 + 1.54i)12-s + (−1.38 + 0.542i)13-s + (−1.26 − 1.06i)14-s + (1.27 + 0.260i)15-s + (0.232 + 0.158i)16-s + (−0.145 + 0.636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00701657 + 0.00259655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00701657 + 0.00259655i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.20 + 1.24i)T \) |
| 7 | \( 1 + (-1.38 + 2.25i)T \) |
| good | 2 | \( 1 + (-0.348 + 2.31i)T + (-1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (2.39 - 1.63i)T + (1.82 - 4.65i)T^{2} \) |
| 11 | \( 1 + (0.509 - 3.38i)T + (-10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (4.98 - 1.95i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (0.598 - 2.62i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 + 3.20iT - 19T^{2} \) |
| 23 | \( 1 + (-2.47 + 8.03i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-2.60 - 8.46i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (4.22 + 2.44i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.00925 - 0.0405i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (7.78 - 5.30i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (4.23 + 2.88i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (4.42 + 0.667i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (0.267 - 0.0610i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (0.860 - 11.4i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.295 - 0.958i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (2.84 - 4.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (10.8 - 2.46i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (2.74 - 2.19i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + (4.95 + 8.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.178 + 0.455i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (6.98 + 8.75i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + (-5.28 + 3.05i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.57855665337604559726664353520, −10.13051550133279177130070852411, −8.573326706324558944765796558138, −7.13807206176847564853079131458, −7.07252907369552268767391509812, −4.80594597265924817160991645478, −4.41668466400819255733078590010, −2.91817546361783166072880425379, −1.72882188930963503737452986396, −0.00471072960627978482897790720,
3.53308446250849930354399844559, 4.83623447406526858354165178068, 5.19782971841251695930287364182, 6.06854870400540919074188426337, 7.40429544771737666636841405539, 8.113457797913475761739555493699, 8.877742143972747560913722134714, 9.806504851899257378695687700844, 11.32355981686605284601346451751, 11.84925641366659796707245555435
