L(s) = 1 | − 10.4·3-s + (17.9 − 52.9i)5-s − 86.7i·7-s − 134.·9-s + 258. i·11-s − 465.·13-s + (−187. + 551. i)15-s + 2.02e3i·17-s + 1.05e3i·19-s + 904. i·21-s − 3.99e3i·23-s + (−2.47e3 − 1.90e3i)25-s + 3.93e3·27-s + 3.24e3i·29-s + 991.·31-s + ⋯ |
L(s) = 1 | − 0.668·3-s + (0.321 − 0.946i)5-s − 0.669i·7-s − 0.552·9-s + 0.644i·11-s − 0.763·13-s + (−0.215 + 0.633i)15-s + 1.70i·17-s + 0.667i·19-s + 0.447i·21-s − 1.57i·23-s + (−0.792 − 0.609i)25-s + 1.03·27-s + 0.715i·29-s + 0.185·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.078685674\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078685674\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-17.9 + 52.9i)T \) |
good | 3 | \( 1 + 10.4T + 243T^{2} \) |
| 7 | \( 1 + 86.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 258. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 465.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.02e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.05e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.99e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.24e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 991.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.77e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.81e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.54e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 4.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.25e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.39e4iT - 8.58e9T^{2} \) |
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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
−10.58920752544919259282429439460, −10.25497091415937063694865953839, −8.911463231436897258450703324796, −8.140617402241796179295660337162, −6.86652248847377915975185362893, −5.85493990591381929390846458197, −4.93018983092607970733997262543, −3.96504698178832432732878747374, −2.11130631221896629197328060309, −0.800509889831658151805998370560,
0.42428013887484635134782145604, 2.36975037305551694391749799927, 3.21234866790597110239998604371, 5.06451554077858606090821763870, 5.74418933754325718542566469953, 6.74627355106662845745434039559, 7.68102430295231265530186842119, 9.069949266918068779212676179872, 9.785571951681569255187872897192, 11.01021394970209124115176744703