L(s) = 1 | + (0.984 − 0.173i)2-s + (1.20 − 1.24i)3-s + (0.939 − 0.342i)4-s + (2.18 − 0.487i)5-s + (0.974 − 1.43i)6-s + (1.28 + 0.739i)7-s + (0.866 − 0.5i)8-s + (−0.0805 − 2.99i)9-s + (2.06 − 0.858i)10-s + (−2.81 + 1.62i)11-s + (0.710 − 1.57i)12-s + (−3.29 − 2.76i)13-s + (1.38 + 0.505i)14-s + (2.03 − 3.29i)15-s + (0.766 − 0.642i)16-s + (0.748 + 4.24i)17-s + ⋯ |
L(s) = 1 | + (0.696 − 0.122i)2-s + (0.697 − 0.716i)3-s + (0.469 − 0.171i)4-s + (0.975 − 0.217i)5-s + (0.397 − 0.584i)6-s + (0.483 + 0.279i)7-s + (0.306 − 0.176i)8-s + (−0.0268 − 0.999i)9-s + (0.652 − 0.271i)10-s + (−0.849 + 0.490i)11-s + (0.205 − 0.455i)12-s + (−0.913 − 0.766i)13-s + (0.371 + 0.135i)14-s + (0.524 − 0.851i)15-s + (0.191 − 0.160i)16-s + (0.181 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.76908 - 1.31641i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.76908 - 1.31641i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (-1.20 + 1.24i)T \) |
| 5 | \( 1 + (-2.18 + 0.487i)T \) |
| 19 | \( 1 + (-1.45 - 4.10i)T \) |
good | 7 | \( 1 + (-1.28 - 0.739i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.81 - 1.62i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.29 + 2.76i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.748 - 4.24i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (7.23 - 2.63i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.791 - 4.48i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.19 - 1.26i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.11T + 37T^{2} \) |
| 41 | \( 1 + (-1.02 + 0.862i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.81 + 7.73i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.06 + 11.6i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (0.503 + 1.38i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.10 - 11.9i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.53 + 3.10i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.31 - 13.1i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.07 - 0.757i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (7.62 + 9.08i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.55 - 5.43i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.29 - 2.23i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.44 + 2.89i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.63 - 9.26i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.28381396973243364066839199910, −10.06340231733054540272312952112, −8.682175438910003520610231751763, −7.902265616451503234677450371776, −7.03370438348567253589377801883, −5.78330128470879967966245618941, −5.27556309753238726691307386634, −3.75290465746985167681752056674, −2.44639951317079163354128832177, −1.71016733396710459916649524073,
2.21918209324773764823956331450, 2.93160635102614624316713970282, 4.45157956813876979671487136525, 5.04019031351365924048269386385, 6.11787044678728180852084820640, 7.33748184351042924463460265288, 8.102640412445505164993053981343, 9.351485645524650970830720241897, 9.899581075581478938513919522588, 10.84456636310399819282370581206