| L(s) = 1 | + 5.44·2-s + 21.6·4-s − 18.8·7-s + 74.0·8-s − 11·11-s − 1.72·13-s − 102.·14-s + 229.·16-s + 3.07·17-s + 123.·19-s − 59.8·22-s − 92.3·23-s − 9.40·26-s − 406.·28-s + 119.·29-s + 292.·31-s + 658.·32-s + 16.7·34-s + 235.·37-s + 673.·38-s + 51.2·41-s + 229.·43-s − 237.·44-s − 502.·46-s − 356.·47-s + 11.3·49-s − 37.3·52-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.70·4-s − 1.01·7-s + 3.27·8-s − 0.301·11-s − 0.0368·13-s − 1.95·14-s + 3.59·16-s + 0.0438·17-s + 1.49·19-s − 0.580·22-s − 0.836·23-s − 0.0709·26-s − 2.74·28-s + 0.764·29-s + 1.69·31-s + 3.63·32-s + 0.0843·34-s + 1.04·37-s + 2.87·38-s + 0.195·41-s + 0.815·43-s − 0.814·44-s − 1.60·46-s − 1.10·47-s + 0.0331·49-s − 0.0996·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.495766029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.495766029\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 5.44T + 8T^{2} \) |
| 7 | \( 1 + 18.8T + 343T^{2} \) |
| 13 | \( 1 + 1.72T + 2.19e3T^{2} \) |
| 17 | \( 1 - 3.07T + 4.91e3T^{2} \) |
| 19 | \( 1 - 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 92.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 51.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 229.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 356.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 117.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 205.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 890.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 655.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 80.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 335.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 432.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 455.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 842.T + 9.12e5T^{2} \) |
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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
−8.238025057124928896754446685785, −7.49554209312207434947717333091, −6.64230043651931998359902760950, −6.15931844285105114748134540148, −5.37124546126558416886954758413, −4.62162957388630439166424448617, −3.74931865153488458426565513341, −3.02218660353829183117683485872, −2.38157429930785291024561248443, −0.972505162021056075507890048055,
0.972505162021056075507890048055, 2.38157429930785291024561248443, 3.02218660353829183117683485872, 3.74931865153488458426565513341, 4.62162957388630439166424448617, 5.37124546126558416886954758413, 6.15931844285105114748134540148, 6.64230043651931998359902760950, 7.49554209312207434947717333091, 8.238025057124928896754446685785
