| L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + 1.75i·5-s + (6.58 + 2.36i)7-s − 2.82i·8-s + (1.23 − 2.14i)10-s + 4.51i·11-s + (−4.80 + 8.32i)13-s + (−6.39 − 7.55i)14-s + (−2.00 + 3.46i)16-s + (0.491 + 0.283i)17-s + (−10.8 − 18.7i)19-s + (−3.03 + 1.75i)20-s + (3.19 − 5.52i)22-s + 27.4i·23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.350i·5-s + (0.941 + 0.337i)7-s − 0.353i·8-s + (0.123 − 0.214i)10-s + 0.410i·11-s + (−0.369 + 0.640i)13-s + (−0.456 − 0.539i)14-s + (−0.125 + 0.216i)16-s + (0.0288 + 0.0166i)17-s + (−0.571 − 0.989i)19-s + (−0.151 + 0.0875i)20-s + (0.145 − 0.251i)22-s + 1.19i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.991856 + 0.615630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991856 + 0.615630i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-6.58 - 2.36i)T \) |
| good | 5 | \( 1 - 1.75iT - 25T^{2} \) |
| 11 | \( 1 - 4.51iT - 121T^{2} \) |
| 13 | \( 1 + (4.80 - 8.32i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-0.491 - 0.283i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (10.8 + 18.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 - 27.4iT - 529T^{2} \) |
| 29 | \( 1 + (48.9 - 28.2i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-19.9 - 34.6i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (7.44 + 12.8i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-28.0 - 16.1i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (1.47 + 2.55i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-47.7 - 27.5i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-48.3 - 27.9i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-10.7 + 6.21i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (39.7 - 68.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (0.142 + 0.246i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 8.92iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-4.02 + 6.97i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (48.8 - 84.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-19.5 + 11.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-47.1 + 27.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-38.0 - 65.8i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.16534018844370660951806234473, −10.57188368455115085634025216141, −9.321643490169549065126660421188, −8.777818690355193294218757660059, −7.54856935742631025239474370426, −6.91299924097165042240407747890, −5.41436321847015341919163270672, −4.27666365207262061458015225858, −2.74037690681254444983949787998, −1.54888827507120949157588878138,
0.66266636206719028502533617754, 2.21230139583523040300541575950, 4.07310790720254024161076813693, 5.23408767580941403574875250677, 6.21172753884855032567188730703, 7.51920220315018668249959986301, 8.152257576590551177704020279608, 8.962721164441321493741416688838, 10.12929940572620042830126516906, 10.79731846624301728525763294336
