L(s) = 1 | + (−2.23 + 0.0113i)5-s + (−0.471 + 0.471i)7-s − 0.335·11-s + (−3.50 + 3.50i)13-s + (−2.53 − 2.53i)17-s + 4.07·19-s + (6.20 − 6.20i)23-s + (4.99 − 0.0506i)25-s − 2.42i·29-s + 6.41·31-s + (1.04 − 1.06i)35-s + (−2.24 − 2.24i)37-s − 5.80i·41-s + (4.87 − 4.87i)43-s + (−1.68 − 1.68i)47-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.00506i)5-s + (−0.178 + 0.178i)7-s − 0.101·11-s + (−0.971 + 0.971i)13-s + (−0.614 − 0.614i)17-s + 0.934·19-s + (1.29 − 1.29i)23-s + (0.999 − 0.0101i)25-s − 0.450i·29-s + 1.15·31-s + (0.177 − 0.179i)35-s + (−0.369 − 0.369i)37-s − 0.906i·41-s + (0.743 − 0.743i)43-s + (−0.245 − 0.245i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.046622681\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.046622681\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 - 0.0113i)T \) |
good | 7 | \( 1 + (0.471 - 0.471i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.335T + 11T^{2} \) |
| 13 | \( 1 + (3.50 - 3.50i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.53 + 2.53i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.07T + 19T^{2} \) |
| 23 | \( 1 + (-6.20 + 6.20i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.42iT - 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 + (2.24 + 2.24i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.80iT - 41T^{2} \) |
| 43 | \( 1 + (-4.87 + 4.87i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.68 + 1.68i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.05 + 3.05i)T + 53iT^{2} \) |
| 59 | \( 1 - 12.2iT - 59T^{2} \) |
| 61 | \( 1 + 7.49iT - 61T^{2} \) |
| 67 | \( 1 + (5.55 + 5.55i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.4iT - 71T^{2} \) |
| 73 | \( 1 + (-5.05 - 5.05i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.85iT - 79T^{2} \) |
| 83 | \( 1 + (4.78 + 4.78i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.33T + 89T^{2} \) |
| 97 | \( 1 + (-10.1 + 10.1i)T - 97iT^{2} \) |
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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
−9.182473854897512530508143767732, −8.790860565280142885899319419733, −7.61380708936630164676495630766, −7.13987394313289288096318866688, −6.30092186483644845479867505030, −4.91547621035403839874831650588, −4.50025451965933692262884491986, −3.26805337757279508331632544600, −2.34953526517910220589976838463, −0.52588005933371739386437334235,
1.02470646862512725554871784638, 2.80809741394883560693584445329, 3.50401094914331657084917031859, 4.66198408825998322657819125595, 5.31742028267413207801099487829, 6.54669547025210494194347240801, 7.39066733346775294408310692619, 7.896270456780886359771361624692, 8.777923549533793452492299559801, 9.695754334223693434109511961661