| L(s) = 1 | + 13.8·5-s − 30.7·7-s − 43.9·11-s − 12.2·13-s − 76.0·17-s + 44.1·19-s − 78.6·23-s + 66.6·25-s + 92.7·29-s + 143.·31-s − 425.·35-s − 32.4·37-s + 335.·41-s + 498.·43-s − 281.·47-s + 599.·49-s + 628.·53-s − 607.·55-s + 504.·59-s + 371.·61-s − 169.·65-s + 162.·67-s + 433.·71-s − 629.·73-s + 1.34e3·77-s − 172.·79-s − 174.·83-s + ⋯ |
| L(s) = 1 | + 1.23·5-s − 1.65·7-s − 1.20·11-s − 0.261·13-s − 1.08·17-s + 0.533·19-s − 0.712·23-s + 0.533·25-s + 0.594·29-s + 0.828·31-s − 2.05·35-s − 0.144·37-s + 1.27·41-s + 1.76·43-s − 0.874·47-s + 1.74·49-s + 1.62·53-s − 1.49·55-s + 1.11·59-s + 0.780·61-s − 0.323·65-s + 0.296·67-s + 0.724·71-s − 1.00·73-s + 1.99·77-s − 0.245·79-s − 0.231·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.588836200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.588836200\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 + 30.7T + 343T^{2} \) |
| 11 | \( 1 + 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 92.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 143.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 628.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 504.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 371.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 162.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 433.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 629.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 172.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 174.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 336.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 84.3T + 9.12e5T^{2} \) |
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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
−9.458364753714543385855944255958, −8.684908288836385399958304489717, −7.53493915166460502041752421523, −6.63476939595013231607871210030, −6.00416098783160082425303248713, −5.29925931328052870897667256034, −4.07998775538846882365237378592, −2.77667357553586694059405334440, −2.31857454177097551538411424767, −0.60149297450461525633310492309,
0.60149297450461525633310492309, 2.31857454177097551538411424767, 2.77667357553586694059405334440, 4.07998775538846882365237378592, 5.29925931328052870897667256034, 6.00416098783160082425303248713, 6.63476939595013231607871210030, 7.53493915166460502041752421523, 8.684908288836385399958304489717, 9.458364753714543385855944255958
