| L(s) = 1 | + 4i·2-s + 12i·3-s − 16·4-s − 48·6-s − 54i·7-s − 64i·8-s + 99·9-s − 121·11-s − 192i·12-s − 540i·13-s + 216·14-s + 256·16-s − 340i·17-s + 396i·18-s + 952·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.769i·3-s − 0.5·4-s − 0.544·6-s − 0.416i·7-s − 0.353i·8-s + 0.407·9-s − 0.301·11-s − 0.384i·12-s − 0.886i·13-s + 0.294·14-s + 0.250·16-s − 0.285i·17-s + 0.288i·18-s + 0.604·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8641998821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8641998821\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 12iT - 243T^{2} \) |
| 7 | \( 1 + 54iT - 1.68e4T^{2} \) |
| 13 | \( 1 + 540iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 340iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 952T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.09e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 62T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.18e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 6.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.33e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.24e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.96e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.75e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.53e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.78e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.89e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 4.19e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.76e4iT - 8.58e9T^{2} \) |
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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
−10.31253500270475538265323791018, −9.658307444966813614261738359810, −8.781620443634072984168757903124, −7.66516076813153424817697957394, −7.10580630192726482356162654192, −5.77306972522642993373537410718, −5.01821732339159486779285812883, −4.05691728203034890120506054659, −3.08176624041432816463495932290, −1.21575568304057233367920266054,
0.20254960858457299953326350271, 1.52190239609972451170481917971, 2.24768482235258438722640924969, 3.56038676407367105927459295097, 4.66937020943132837263732695714, 5.79142373792431701101718990507, 6.87578070280878620209866756323, 7.69920203828220826373517171852, 8.734494200959694695819037840341, 9.520673380735974292840527445597
