L(s) = 1 | − 2.64i·3-s − 3.46i·5-s + i·7-s − 4.00·9-s − 2.64·13-s − 9.16·15-s − 3·17-s + (3.46 − 2.64i)19-s + 2.64·21-s + 3i·23-s − 6.99·25-s + 2.64i·27-s − 7.93·29-s + 3.46·35-s − 10.5·37-s + ⋯ |
L(s) = 1 | − 1.52i·3-s − 1.54i·5-s + 0.377i·7-s − 1.33·9-s − 0.733·13-s − 2.36·15-s − 0.727·17-s + (0.794 − 0.606i)19-s + 0.577·21-s + 0.625i·23-s − 1.39·25-s + 0.509i·27-s − 1.47·29-s + 0.585·35-s − 1.73·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9912162384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9912162384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.46 + 2.64i)T \) |
good | 3 | \( 1 + 2.64iT - 3T^{2} \) |
| 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 9.16iT - 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 + 7.93iT - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 13.2iT - 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 9.16iT - 89T^{2} \) |
| 97 | \( 1 + 9.16iT - 97T^{2} \) |
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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
−8.920400238031959783857070502765, −8.535736412729138544380388227081, −7.38832715214300264129117747147, −7.10455797348495767117272283345, −5.67424687302737403771903851632, −5.30948710148849354762320705937, −4.06221029139059275431661223399, −2.45215125955196814113054812171, −1.56894531635064829727391448516, −0.41008454955637129529280685348,
2.36995107046449155993594136316, 3.36092327294696662444041609657, 4.01017026274360063816103681163, 5.01668522586578120753687909190, 5.96634771595020563547677806788, 6.98938351456236038356733548547, 7.62992885962810927167648630040, 8.880587432476462547660179799713, 9.678791840528925321424495299716, 10.24508049914096377918018196435