L(s) = 1 | + (−1.41 + 1.41i)3-s + (−1.94 + 1.09i)5-s + (0.0496 − 2.64i)7-s − 0.987i·9-s − 5.75·11-s + (2.89 − 2.89i)13-s + (1.20 − 4.29i)15-s + (−2.13 − 2.13i)17-s − 2.18·19-s + (3.66 + 3.80i)21-s + (−4.79 − 4.79i)23-s + (2.60 − 4.26i)25-s + (−2.84 − 2.84i)27-s + 5.19i·29-s + 6.68i·31-s + ⋯ |
L(s) = 1 | + (−0.815 + 0.815i)3-s + (−0.871 + 0.489i)5-s + (0.0187 − 0.999i)7-s − 0.329i·9-s − 1.73·11-s + (0.802 − 0.802i)13-s + (0.311 − 1.10i)15-s + (−0.517 − 0.517i)17-s − 0.502·19-s + (0.799 + 0.830i)21-s + (−1.00 − 1.00i)23-s + (0.520 − 0.853i)25-s + (−0.546 − 0.546i)27-s + 0.964i·29-s + 1.20i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.748 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0168702 - 0.0445088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0168702 - 0.0445088i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.94 - 1.09i)T \) |
| 7 | \( 1 + (-0.0496 + 2.64i)T \) |
good | 3 | \( 1 + (1.41 - 1.41i)T - 3iT^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + (-2.89 + 2.89i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.13 + 2.13i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 + (4.79 + 4.79i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.19iT - 29T^{2} \) |
| 31 | \( 1 - 6.68iT - 31T^{2} \) |
| 37 | \( 1 + (6.50 - 6.50i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.846iT - 41T^{2} \) |
| 43 | \( 1 + (-2.68 - 2.68i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.55 - 4.55i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.750 + 0.750i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.25T + 59T^{2} \) |
| 61 | \( 1 + 7.62iT - 61T^{2} \) |
| 67 | \( 1 + (2.14 - 2.14i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 + (3.51 - 3.51i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.15iT - 79T^{2} \) |
| 83 | \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 + (5.66 + 5.66i)T + 97iT^{2} \) |
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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
−11.04949160991567521620125135467, −10.61791490317873371251852640167, −10.23347577262910927135985892735, −8.417409201119129163678913995788, −7.65091067065659845287802162882, −6.49854987826942521207134934496, −5.15918749170235550055858932706, −4.32793793205489409840484238164, −3.07083016647835247001640975067, −0.03765006726445107881774389431,
2.07738676086986707568705211653, 3.98482226117003982947906197794, 5.41689693581734580349201148487, 6.11509550358515582101113535132, 7.42744172498779289589939973700, 8.233708300123446433594721710284, 9.178748011329688456279561277326, 10.69116906123541306627251045336, 11.52838922193028308356336962351, 12.14510176548534363576818314240