| L(s) = 1 | − 1.41i·2-s + (−0.554 + 2.94i)3-s − 2.00·4-s + (4.16 + 0.783i)6-s + 2.64·7-s + 2.82i·8-s + (−8.38 − 3.26i)9-s − 2.58i·11-s + (1.10 − 5.89i)12-s − 0.180·13-s − 3.74i·14-s + 4.00·16-s − 7.25i·17-s + (−4.62 + 11.8i)18-s + 21.4·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.184 + 0.982i)3-s − 0.500·4-s + (0.694 + 0.130i)6-s + 0.377·7-s + 0.353i·8-s + (−0.931 − 0.363i)9-s − 0.234i·11-s + (0.0923 − 0.491i)12-s − 0.0138·13-s − 0.267i·14-s + 0.250·16-s − 0.426i·17-s + (−0.256 + 0.658i)18-s + 1.12·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.632977032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632977032\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (0.554 - 2.94i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 11 | \( 1 + 2.58iT - 121T^{2} \) |
| 13 | \( 1 + 0.180T + 169T^{2} \) |
| 17 | \( 1 + 7.25iT - 289T^{2} \) |
| 19 | \( 1 - 21.4T + 361T^{2} \) |
| 23 | \( 1 - 11.8iT - 529T^{2} \) |
| 29 | \( 1 + 29.5iT - 841T^{2} \) |
| 31 | \( 1 + 49.5T + 961T^{2} \) |
| 37 | \( 1 - 43.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 20.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 7.05iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 55.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 100. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.44T + 3.72e3T^{2} \) |
| 67 | \( 1 - 85.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 97.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 17.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 3.08iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 80.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 105.T + 9.40e3T^{2} \) |
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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
−9.632250947156805454052948634189, −9.280621260059162487108087083526, −8.253560336639344429791384084935, −7.36056845974139384378655162278, −5.89256607093019190019719593043, −5.25394812216225873213201842887, −4.28333052305307972181872005128, −3.46815767047823313749337336176, −2.41227366467757849123642939306, −0.78402015853705361926283933020,
0.823507998126972648529275712365, 2.09151034684958903366157864822, 3.48192288145639748459072464966, 4.82411330254773042230890448802, 5.62434456617978702633470798213, 6.41602160236650867657561180202, 7.38780920133407521777624886052, 7.74940352778249106723168573101, 8.760163446203762861549830196885, 9.446983726685847371149949647334
