| L(s) = 1 | + (−2.64 + 1.41i)3-s − 7.48i·5-s + 5.29·7-s + (5 − 7.48i)9-s − 14.1i·11-s + 10·13-s + (10.5 + 19.7i)15-s − 26.4·19-s + (−14.0 + 7.48i)21-s − 16.9i·23-s − 31·25-s + (−2.64 + 26.8i)27-s + 37.4i·29-s + 26.4·31-s + (20.0 + 37.4i)33-s + ⋯ |
| L(s) = 1 | + (−0.881 + 0.471i)3-s − 1.49i·5-s + 0.755·7-s + (0.555 − 0.831i)9-s − 1.28i·11-s + 0.769·13-s + (0.705 + 1.31i)15-s − 1.39·19-s + (−0.666 + 0.356i)21-s − 0.737i·23-s − 1.23·25-s + (−0.0979 + 0.995i)27-s + 1.29i·29-s + 0.853·31-s + (0.606 + 1.13i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.876893 - 0.525584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.876893 - 0.525584i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.64 - 1.41i)T \) |
| good | 5 | \( 1 + 7.48iT - 25T^{2} \) |
| 7 | \( 1 - 5.29T + 49T^{2} \) |
| 11 | \( 1 + 14.1iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 26.4T + 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 - 37.4iT - 841T^{2} \) |
| 31 | \( 1 - 26.4T + 961T^{2} \) |
| 37 | \( 1 - 10T + 1.36e3T^{2} \) |
| 41 | \( 1 - 14.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 58.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 37.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 98.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90T + 3.72e3T^{2} \) |
| 67 | \( 1 + 5.29T + 4.48e3T^{2} \) |
| 71 | \( 1 + 28.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30T + 5.32e3T^{2} \) |
| 79 | \( 1 - 26.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 25.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 74.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 10T + 9.40e3T^{2} \) |
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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
−13.29657288636785719545380600900, −12.43005051137653048915225402878, −11.36975774812195344889822496593, −10.55291489830557503628181039261, −8.947483781057464908053361423468, −8.338575705543091190947850073912, −6.26404146557412019274676740834, −5.18149764702616660346862318318, −4.14808209795628261790621118818, −0.965923133741797720824814046479,
2.11908932813620542016928336159, 4.36018741684057147323911890198, 6.03493727108382524964382299464, 6.97629746554048834352943675060, 7.968327221503939783166704706975, 9.974561427876292750896703519481, 10.89309766522586691975761754398, 11.55401501931313713929388151169, 12.75398286739697630814034162232, 13.93299818195554555302828228983
