L(s) = 1 | + 18.5·2-s + 248.·3-s − 166.·4-s + 410.·5-s + 4.61e3·6-s − 5.15e3·7-s − 1.26e4·8-s + 4.19e4·9-s + 7.62e3·10-s − 3.60e4·11-s − 4.12e4·12-s + 1.26e5·13-s − 9.58e4·14-s + 1.01e5·15-s − 1.49e5·16-s + 7.79e5·18-s − 4.12e5·19-s − 6.82e4·20-s − 1.27e6·21-s − 6.70e5·22-s − 2.06e6·23-s − 3.13e6·24-s − 1.78e6·25-s + 2.34e6·26-s + 5.51e6·27-s + 8.57e5·28-s + 6.64e6·29-s + ⋯ |
L(s) = 1 | + 0.821·2-s + 1.76·3-s − 0.324·4-s + 0.293·5-s + 1.45·6-s − 0.811·7-s − 1.08·8-s + 2.12·9-s + 0.241·10-s − 0.742·11-s − 0.574·12-s + 1.22·13-s − 0.666·14-s + 0.519·15-s − 0.569·16-s + 1.74·18-s − 0.726·19-s − 0.0953·20-s − 1.43·21-s − 0.610·22-s − 1.53·23-s − 1.92·24-s − 0.913·25-s + 1.00·26-s + 1.99·27-s + 0.263·28-s + 1.74·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 18.5T + 512T^{2} \) |
| 3 | \( 1 - 248.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 410.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.15e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.60e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.26e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 4.12e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.06e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.64e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.55e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.09e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.24e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.86e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.45e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.58e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.76e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.94e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.59e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.57e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.78e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.91e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.16e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.16e9T + 7.60e17T^{2} \) |
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
−9.712169245555303637444758209587, −8.574996581354149744839970737571, −8.270313967202470689791615308228, −6.75801958792883155153991380209, −5.76050029600979339442253617297, −4.32050917839735579841390555402, −3.56970427283641424558112413891, −2.83908134835923674740030567101, −1.76207790587087321662559535873, 0,
1.76207790587087321662559535873, 2.83908134835923674740030567101, 3.56970427283641424558112413891, 4.32050917839735579841390555402, 5.76050029600979339442253617297, 6.75801958792883155153991380209, 8.270313967202470689791615308228, 8.574996581354149744839970737571, 9.712169245555303637444758209587