L(s) = 1 | + 25.0i·3-s + 25i·5-s − 103.·7-s − 384.·9-s − 740. i·11-s − 892. i·13-s − 626.·15-s + 1.13e3·17-s + 1.15e3i·19-s − 2.59e3i·21-s − 1.60e3·23-s − 625·25-s − 3.54e3i·27-s + 2.15e3i·29-s − 4.95e3·31-s + ⋯ |
L(s) = 1 | + 1.60i·3-s + 0.447i·5-s − 0.799·7-s − 1.58·9-s − 1.84i·11-s − 1.46i·13-s − 0.718·15-s + 0.955·17-s + 0.734i·19-s − 1.28i·21-s − 0.631·23-s − 0.200·25-s − 0.936i·27-s + 0.476i·29-s − 0.926·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7948737247\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7948737247\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 25iT \) |
good | 3 | \( 1 - 25.0iT - 243T^{2} \) |
| 7 | \( 1 + 103.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 740. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 892. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.15e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.15e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.40e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 3.78e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.30e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.00e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.43e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.20e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.82e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.49e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.39e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.41e4T + 8.58e9T^{2} \) |
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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
−11.52162732307164379903128968669, −10.49803987293822008225888022150, −10.12421724144372631992936191065, −8.958066292785603760530704410175, −7.929791807972240720070347364579, −6.05066734353847133311865873473, −5.37635300846805930602325236200, −3.50414458465136102847751779713, −3.27140124774868378115605725140, −0.26830791471723154911605044202,
1.35836837350507643172736036392, 2.38961404938964920731693102829, 4.34889239831630825474061988152, 5.97225907661297231886929476819, 7.02347936398138485380125186956, 7.56334117771053188897486134911, 9.024702440665294873365977879974, 9.876097290830951109107246269058, 11.59970055168734669492484450112, 12.37031984570382866855603181347