| L(s) = 1 | + 2·2-s + 9.15·3-s + 4·4-s + 18.3·6-s + 8·8-s + 56.7·9-s − 35.7·11-s + 36.6·12-s + 67.4·13-s + 16·16-s − 19.1·17-s + 113.·18-s + 86.6·19-s − 71.5·22-s − 195.·23-s + 73.2·24-s + 134.·26-s + 272.·27-s + 272.·29-s − 21.6·31-s + 32·32-s − 327.·33-s − 38.3·34-s + 227.·36-s − 132.·37-s + 173.·38-s + 617.·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.76·3-s + 0.5·4-s + 1.24·6-s + 0.353·8-s + 2.10·9-s − 0.980·11-s + 0.880·12-s + 1.43·13-s + 0.250·16-s − 0.273·17-s + 1.48·18-s + 1.04·19-s − 0.693·22-s − 1.76·23-s + 0.622·24-s + 1.01·26-s + 1.94·27-s + 1.74·29-s − 0.125·31-s + 0.176·32-s − 1.72·33-s − 0.193·34-s + 1.05·36-s − 0.589·37-s + 0.739·38-s + 2.53·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.119997533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.119997533\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 9.15T + 27T^{2} \) |
| 11 | \( 1 + 35.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 21.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 67.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 107.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 609.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 645.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 140.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 834.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 491.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 479.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 610.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 153.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 784.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 227.T + 9.12e5T^{2} \) |
| 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
−8.516518034139553370134228477022, −7.87819101150490422710895809210, −7.27134274793594105545128925930, −6.26359188162475253079523053182, −5.39128544902821324798474102290, −4.23619184706375248501710678050, −3.71446746885510555592058397356, −2.83755031320259046990914659230, −2.21201117569832499318439089985, −1.10004779568060768161693051758,
1.10004779568060768161693051758, 2.21201117569832499318439089985, 2.83755031320259046990914659230, 3.71446746885510555592058397356, 4.23619184706375248501710678050, 5.39128544902821324798474102290, 6.26359188162475253079523053182, 7.27134274793594105545128925930, 7.87819101150490422710895809210, 8.516518034139553370134228477022
