| L(s) = 1 | + 1.61·2-s − 5.39·4-s + 1.20·5-s − 28.2·7-s − 21.6·8-s + 1.95·10-s − 31.7·11-s − 45.6·14-s + 8.22·16-s + 16.0·17-s + 58.5·19-s − 6.51·20-s − 51.3·22-s − 152.·23-s − 123.·25-s + 152.·28-s − 265.·29-s + 56.9·31-s + 186.·32-s + 25.9·34-s − 34.1·35-s − 444.·37-s + 94.5·38-s − 26.1·40-s − 189.·41-s − 132.·43-s + 171.·44-s + ⋯ |
| L(s) = 1 | + 0.570·2-s − 0.674·4-s + 0.108·5-s − 1.52·7-s − 0.955·8-s + 0.0617·10-s − 0.871·11-s − 0.871·14-s + 0.128·16-s + 0.229·17-s + 0.706·19-s − 0.0728·20-s − 0.497·22-s − 1.38·23-s − 0.988·25-s + 1.02·28-s − 1.69·29-s + 0.329·31-s + 1.02·32-s + 0.131·34-s − 0.165·35-s − 1.97·37-s + 0.403·38-s − 0.103·40-s − 0.720·41-s − 0.470·43-s + 0.587·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6050016132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6050016132\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 8T^{2} \) |
| 5 | \( 1 - 1.20T + 125T^{2} \) |
| 7 | \( 1 + 28.2T + 343T^{2} \) |
| 11 | \( 1 + 31.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 16.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 56.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 444.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 189.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 513.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 619.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 597.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 332.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 679.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 88.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 154.T + 9.12e5T^{2} \) |
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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
−9.282296856128329108157590926089, −8.335016687405434624203808314269, −7.46612035084573169905795082664, −6.44062079281473150258233568451, −5.69013968752473797844312485999, −5.09509783139135184257151247916, −3.71690694843485124345260324759, −3.43929891005830324417984151888, −2.16995886977924429517591279566, −0.32760084555205158362337677265,
0.32760084555205158362337677265, 2.16995886977924429517591279566, 3.43929891005830324417984151888, 3.71690694843485124345260324759, 5.09509783139135184257151247916, 5.69013968752473797844312485999, 6.44062079281473150258233568451, 7.46612035084573169905795082664, 8.335016687405434624203808314269, 9.282296856128329108157590926089
