L(s) = 1 | + 0.414i·3-s + (−2.12 − 0.707i)5-s + 0.414i·7-s + 2.82·9-s − 1.41·11-s + 3.82i·13-s + (0.292 − 0.878i)15-s + i·17-s + 19-s − 0.171·21-s + 3.24i·23-s + (3.99 + 3i)25-s + 2.41i·27-s + 1.82·29-s + 0.585·31-s + ⋯ |
L(s) = 1 | + 0.239i·3-s + (−0.948 − 0.316i)5-s + 0.156i·7-s + 0.942·9-s − 0.426·11-s + 1.06i·13-s + (0.0756 − 0.226i)15-s + 0.242i·17-s + 0.229·19-s − 0.0374·21-s + 0.676i·23-s + (0.799 + 0.600i)25-s + 0.464i·27-s + 0.339·29-s + 0.105·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.960873 + 0.692558i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.960873 + 0.692558i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.12 + 0.707i)T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.414iT - 3T^{2} \) |
| 7 | \( 1 - 0.414iT - 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 3.82iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 23 | \( 1 - 3.24iT - 23T^{2} \) |
| 29 | \( 1 - 1.82T + 29T^{2} \) |
| 31 | \( 1 - 0.585T + 31T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 6.24iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 3.82iT - 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 0.585T + 61T^{2} \) |
| 67 | \( 1 - 8.07iT - 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 8.17iT - 73T^{2} \) |
| 79 | \( 1 + 4.82T + 79T^{2} \) |
| 83 | \( 1 + 14.4iT - 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 3.65iT - 97T^{2} \) |
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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
−10.50631839468295420561996499722, −9.605734905221495732950580697189, −8.853222600977272364494905173075, −7.84814245473558443850083954746, −7.21294797274446815649898991160, −6.15065904734731538771193300256, −4.78275441402741845589119708664, −4.26823320588900387482720975867, −3.10069969980703888622577382641, −1.42128487423258976484156251315,
0.66784810928231549694847414130, 2.50766382639485264894953352878, 3.68108591139365277382034526585, 4.58791245096905576973691042751, 5.73070765141885938716729093325, 6.95903627960470791402390147268, 7.52484370215071357498131296285, 8.230411374971702330389934704258, 9.309944443354267417584833079377, 10.53593484341264417916507489587