| L(s) = 1 | + 5·5-s + 5.60·7-s + 53.5·11-s + 29.6·13-s + 109.·17-s + 67.6·19-s − 87.4·23-s + 25·25-s − 141.·29-s + 219.·31-s + 28.0·35-s + 436.·37-s − 68.7·41-s − 16.9·43-s + 566.·47-s − 311.·49-s − 95.0·53-s + 267.·55-s + 306.·59-s − 744.·61-s + 148.·65-s − 521.·67-s − 678.·71-s − 879.·73-s + 300.·77-s + 698.·79-s − 1.19e3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.302·7-s + 1.46·11-s + 0.633·13-s + 1.55·17-s + 0.816·19-s − 0.792·23-s + 0.200·25-s − 0.909·29-s + 1.26·31-s + 0.135·35-s + 1.93·37-s − 0.261·41-s − 0.0600·43-s + 1.75·47-s − 0.908·49-s − 0.246·53-s + 0.656·55-s + 0.676·59-s − 1.56·61-s + 0.283·65-s − 0.950·67-s − 1.13·71-s − 1.41·73-s + 0.444·77-s + 0.994·79-s − 1.57·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.497869698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.497869698\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| good | 7 | \( 1 - 5.60T + 343T^{2} \) |
| 11 | \( 1 - 53.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 87.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 436.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 68.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 16.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 566.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 95.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 306.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 744.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 521.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 678.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 879.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 698.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 840.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−8.798741739144260061020595890146, −7.907188063133290343527553725857, −7.25593137476536045421016334869, −6.06687952113027637796009514626, −5.88130746269755240157892364941, −4.63368401722123216757467206939, −3.81460803817082611539188554412, −2.92084131241598527841500768769, −1.57633405942333002231355928279, −0.958356064454058177720609227916,
0.958356064454058177720609227916, 1.57633405942333002231355928279, 2.92084131241598527841500768769, 3.81460803817082611539188554412, 4.63368401722123216757467206939, 5.88130746269755240157892364941, 6.06687952113027637796009514626, 7.25593137476536045421016334869, 7.907188063133290343527553725857, 8.798741739144260061020595890146
