| L(s) = 1 | + 2·2-s − 1.86·3-s + 4·4-s + 5.44·5-s − 3.72·6-s − 7·7-s + 8·8-s − 23.5·9-s + 10.8·10-s − 29.9·11-s − 7.44·12-s − 14·14-s − 10.1·15-s + 16·16-s + 112.·17-s − 47.0·18-s − 50.2·19-s + 21.7·20-s + 13.0·21-s − 59.8·22-s − 31.0·23-s − 14.8·24-s − 95.3·25-s + 94.0·27-s − 28·28-s + 150.·29-s − 20.2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.358·3-s + 0.5·4-s + 0.487·5-s − 0.253·6-s − 0.377·7-s + 0.353·8-s − 0.871·9-s + 0.344·10-s − 0.820·11-s − 0.179·12-s − 0.267·14-s − 0.174·15-s + 0.250·16-s + 1.60·17-s − 0.616·18-s − 0.606·19-s + 0.243·20-s + 0.135·21-s − 0.579·22-s − 0.281·23-s − 0.126·24-s − 0.762·25-s + 0.670·27-s − 0.188·28-s + 0.962·29-s − 0.123·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.604374824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.604374824\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.86T + 27T^{2} \) |
| 5 | \( 1 - 5.44T + 125T^{2} \) |
| 11 | \( 1 + 29.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 31.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 150.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 49.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 269.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 499.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 161.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 473.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 545.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 270.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 941.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 770.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 342.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 294.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 826.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 96.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 716.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.369221345885074270106751416844, −7.934298385512456329177776241184, −6.80243800026850349623328855225, −6.09745582179260394905822564869, −5.49978725313792793792117061960, −4.89900185401571555107228676262, −3.66843726214613718496899070060, −2.91591601949143199389227931882, −2.02169436448690118046791076489, −0.63281971083060097349711633691,
0.63281971083060097349711633691, 2.02169436448690118046791076489, 2.91591601949143199389227931882, 3.66843726214613718496899070060, 4.89900185401571555107228676262, 5.49978725313792793792117061960, 6.09745582179260394905822564869, 6.80243800026850349623328855225, 7.934298385512456329177776241184, 8.369221345885074270106751416844
