L(s) = 1 | + 2.41i·2-s − 1.41·3-s − 3.82·4-s − i·5-s − 3.41i·6-s + 4.82i·7-s − 4.41i·8-s − 0.999·9-s + 2.41·10-s + 3.41i·11-s + 5.41·12-s − 11.6·14-s + 1.41i·15-s + 2.99·16-s − 0.828·17-s − 2.41i·18-s + ⋯ |
L(s) = 1 | + 1.70i·2-s − 0.816·3-s − 1.91·4-s − 0.447i·5-s − 1.39i·6-s + 1.82i·7-s − 1.56i·8-s − 0.333·9-s + 0.763·10-s + 1.02i·11-s + 1.56·12-s − 3.11·14-s + 0.365i·15-s + 0.749·16-s − 0.200·17-s − 0.569i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.281708 - 0.150765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.281708 - 0.150765i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.41iT - 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 11 | \( 1 - 3.41iT - 11T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 0.585iT - 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 + 1.75iT - 31T^{2} \) |
| 37 | \( 1 + 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 3.17iT - 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 4.82iT - 47T^{2} \) |
| 53 | \( 1 - 2.48T + 53T^{2} \) |
| 59 | \( 1 - 1.75iT - 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 3.17iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 7.65iT - 97T^{2} \) |
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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
−10.96184277883929763509152715492, −9.483293762323588189663094309274, −9.082532639963212806809024666230, −8.274675599832729048513779208471, −7.38787876365808593939091285921, −6.37310602089423954763401098121, −5.65786232687484104842294795592, −5.29475780043167250898893301307, −4.31221006717434068661731417328, −2.32128960413340616336943002632,
0.18910554330363969961057759617, 1.26387986542974038084162904967, 2.90924621206307939805658916047, 3.75724676494181573123005356296, 4.56861212672698714531854720279, 5.79294103948924174409063620893, 6.82221162198251592563136398903, 7.895747582404025460684473351989, 8.993494186611507785977355054277, 10.02498869664391126552322921650