L(s) = 1 | + (0.659 + 1.14i)2-s + (0.130 − 0.226i)4-s + (0.729 + 2.11i)5-s + (−2.12 + 1.57i)7-s + 2.98·8-s + (−1.93 + 2.22i)10-s + (2.08 + 1.20i)11-s − 1.69·13-s + (−3.19 − 1.38i)14-s + (1.70 + 2.95i)16-s + (−0.831 − 0.480i)17-s + (3.56 − 2.05i)19-s + (0.574 + 0.111i)20-s + 3.17i·22-s + (2.88 + 4.99i)23-s + ⋯ |
L(s) = 1 | + (0.466 + 0.807i)2-s + (0.0654 − 0.113i)4-s + (0.326 + 0.945i)5-s + (−0.803 + 0.595i)7-s + 1.05·8-s + (−0.610 + 0.704i)10-s + (0.629 + 0.363i)11-s − 0.469·13-s + (−0.855 − 0.371i)14-s + (0.425 + 0.737i)16-s + (−0.201 − 0.116i)17-s + (0.818 − 0.472i)19-s + (0.128 + 0.0248i)20-s + 0.677i·22-s + (0.601 + 1.04i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0424 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0424 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29366 + 1.23986i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29366 + 1.23986i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.729 - 2.11i)T \) |
| 7 | \( 1 + (2.12 - 1.57i)T \) |
good | 2 | \( 1 + (-0.659 - 1.14i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.08 - 1.20i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 + (0.831 + 0.480i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.56 + 2.05i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.88 - 4.99i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.56iT - 29T^{2} \) |
| 31 | \( 1 + (7.58 + 4.37i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.44 + 1.98i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + (8.75 - 5.05i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.35 + 11.0i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.38 - 5.86i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.98 + 1.14i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.15 + 2.39i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.24iT - 71T^{2} \) |
| 73 | \( 1 + (-7.34 + 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.892 - 1.54i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.62iT - 83T^{2} \) |
| 89 | \( 1 + (0.220 + 0.382i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−11.82696542044774384052701244347, −11.01575356093756174358778654527, −9.836175044335833946662356551166, −9.321654131808812127390265819419, −7.56525160640978410156360116999, −6.92387219755237639683233700781, −6.06841457386007965063765567447, −5.23152524864495444850102352461, −3.67135556284920355103555665447, −2.21931899625330541728950572378,
1.32403804819933185887476958276, 3.00441774760159485551886699822, 4.06892980091378762799563735818, 5.11301133473675419857213198601, 6.51730974734709937070543502234, 7.57471702544286014823947858757, 8.802859592554343244136516495623, 9.717404329568350013901994126622, 10.61980104670706628671401566003, 11.58221793366150700272227835498