L(s) = 1 | + 2-s − 1.61·3-s + 4-s − 1.61·6-s + 8-s + 1.61·9-s − 1.61·12-s + 16-s + 0.618·17-s + 1.61·18-s + 0.618·19-s − 1.61·24-s + 25-s − 27-s + 32-s + 0.618·34-s + 1.61·36-s + 0.618·38-s − 1.61·41-s − 1.61·43-s − 1.61·48-s + 49-s + 50-s − 1.00·51-s − 54-s − 1.00·57-s + 0.618·59-s + ⋯ |
L(s) = 1 | + 2-s − 1.61·3-s + 4-s − 1.61·6-s + 8-s + 1.61·9-s − 1.61·12-s + 16-s + 0.618·17-s + 1.61·18-s + 0.618·19-s − 1.61·24-s + 25-s − 27-s + 32-s + 0.618·34-s + 1.61·36-s + 0.618·38-s − 1.61·41-s − 1.61·43-s − 1.61·48-s + 49-s + 50-s − 1.00·51-s − 54-s − 1.00·57-s + 0.618·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.211688611\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.211688611\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 1.61T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 0.618T + T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 0.618T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.61T + T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−10.45308004530249013086212206855, −9.900695998176153561029883690998, −8.359499259682911007594734363120, −7.13496463655196913782216692557, −6.68064557644438698867877027526, −5.62978688784288487761613360424, −5.21284685229712873470977647786, −4.28217600124830721024411176975, −3.09005058543264379774918784845, −1.38732321998929055135039663178,
1.38732321998929055135039663178, 3.09005058543264379774918784845, 4.28217600124830721024411176975, 5.21284685229712873470977647786, 5.62978688784288487761613360424, 6.68064557644438698867877027526, 7.13496463655196913782216692557, 8.359499259682911007594734363120, 9.900695998176153561029883690998, 10.45308004530249013086212206855