| L(s) = 1 | + i·2-s − 4-s − 4.29·5-s − i·8-s − 4.29i·10-s + 5.20i·11-s + 2.81i·13-s + 16-s + 4.02·17-s − 1.77i·19-s + 4.29·20-s − 5.20·22-s + 7.70i·23-s + 13.4·25-s − 2.81·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.92·5-s − 0.353i·8-s − 1.35i·10-s + 1.56i·11-s + 0.780i·13-s + 0.250·16-s + 0.976·17-s − 0.408i·19-s + 0.960·20-s − 1.10·22-s + 1.60i·23-s + 2.69·25-s − 0.551·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3247618231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3247618231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.29T + 5T^{2} \) |
| 11 | \( 1 - 5.20iT - 11T^{2} \) |
| 13 | \( 1 - 2.81iT - 13T^{2} \) |
| 17 | \( 1 - 4.02T + 17T^{2} \) |
| 19 | \( 1 + 1.77iT - 19T^{2} \) |
| 23 | \( 1 - 7.70iT - 23T^{2} \) |
| 29 | \( 1 - 5.02iT - 29T^{2} \) |
| 31 | \( 1 + 1.25iT - 31T^{2} \) |
| 37 | \( 1 + 9.17T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 + 7.59T + 47T^{2} \) |
| 53 | \( 1 + 0.167iT - 53T^{2} \) |
| 59 | \( 1 + 5.99T + 59T^{2} \) |
| 61 | \( 1 + 7.79iT - 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 0.539iT - 71T^{2} \) |
| 73 | \( 1 + 3.80iT - 73T^{2} \) |
| 79 | \( 1 - 1.99T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 1.29T + 89T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
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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
−9.265454899428869187227101087557, −8.354662960437887042054930438817, −7.72613389795235802728418025782, −7.15135925716175214527327758080, −6.75585661852615415690202995715, −5.29069621066136925408779853359, −4.70625562305641812455501932990, −3.90834630976022805697342239416, −3.24417854079944095789598682318, −1.52906794429607114897296795673,
0.14166111798285187218648087728, 0.973369839164618990396853668811, 2.84360267451802294802222826000, 3.40789216678162974498442370307, 4.06128556660057093228027273909, 4.99079264397719878981583581307, 5.91122662763549826300405849155, 6.96611161632512111040450526807, 8.000585281492632146043165483968, 8.233947790056256683513899008053
