| L(s) = 1 | − 0.267·2-s + 4.19·3-s − 7.92·4-s − 4.12·5-s − 1.12·6-s + 9.26·7-s + 4.26·8-s − 9.39·9-s + 1.10·10-s + 2.87·11-s − 33.2·12-s + 51.3·13-s − 2.48·14-s − 17.3·15-s + 62.2·16-s − 97.0·17-s + 2.51·18-s + 89.8·19-s + 32.6·20-s + 38.8·21-s − 0.770·22-s + 17.9·24-s − 107.·25-s − 13.7·26-s − 152.·27-s − 73.4·28-s − 194.·29-s + ⋯ |
| L(s) = 1 | − 0.0947·2-s + 0.807·3-s − 0.991·4-s − 0.368·5-s − 0.0765·6-s + 0.500·7-s + 0.188·8-s − 0.347·9-s + 0.0349·10-s + 0.0788·11-s − 0.800·12-s + 1.09·13-s − 0.0474·14-s − 0.297·15-s + 0.973·16-s − 1.38·17-s + 0.0329·18-s + 1.08·19-s + 0.365·20-s + 0.404·21-s − 0.00746·22-s + 0.152·24-s − 0.863·25-s − 0.103·26-s − 1.08·27-s − 0.495·28-s − 1.24·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + 0.267T + 8T^{2} \) |
| 3 | \( 1 - 4.19T + 27T^{2} \) |
| 5 | \( 1 + 4.12T + 125T^{2} \) |
| 7 | \( 1 - 9.26T + 343T^{2} \) |
| 11 | \( 1 - 2.87T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 89.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 23.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 334.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 290.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 681.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 814.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 513.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 660.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 447.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 864.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 390.T + 9.12e5T^{2} \) |
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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
−9.568771795061561506313682611142, −9.136609959410572516232461787681, −8.160640705044009513444323313412, −7.81372111262827861094583859727, −6.26136903590134623644038494051, −5.11735994326322244027496324253, −4.06266092931603788389343627597, −3.24028782975744983405208301317, −1.64545778910554448954474261357, 0,
1.64545778910554448954474261357, 3.24028782975744983405208301317, 4.06266092931603788389343627597, 5.11735994326322244027496324253, 6.26136903590134623644038494051, 7.81372111262827861094583859727, 8.160640705044009513444323313412, 9.136609959410572516232461787681, 9.568771795061561506313682611142
