| L(s) = 1 | + 11.8·2-s + 67.6·3-s − 372.·4-s − 2.39e3·5-s + 799.·6-s + 1.13e4·7-s − 1.04e4·8-s − 1.51e4·9-s − 2.82e4·10-s + 1.77e4·11-s − 2.51e4·12-s − 7.61e4·13-s + 1.34e5·14-s − 1.61e5·15-s + 6.70e4·16-s − 1.78e5·18-s + 6.61e5·19-s + 8.89e5·20-s + 7.68e5·21-s + 2.09e5·22-s + 1.64e6·23-s − 7.07e5·24-s + 3.76e6·25-s − 8.99e5·26-s − 2.35e6·27-s − 4.22e6·28-s − 1.49e6·29-s + ⋯ |
| L(s) = 1 | + 0.522·2-s + 0.482·3-s − 0.727·4-s − 1.71·5-s + 0.251·6-s + 1.78·7-s − 0.902·8-s − 0.767·9-s − 0.893·10-s + 0.365·11-s − 0.350·12-s − 0.739·13-s + 0.933·14-s − 0.825·15-s + 0.255·16-s − 0.400·18-s + 1.16·19-s + 1.24·20-s + 0.862·21-s + 0.190·22-s + 1.22·23-s − 0.435·24-s + 1.92·25-s − 0.386·26-s − 0.852·27-s − 1.29·28-s − 0.393·29-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 | \( 1 - 11.8T + 512T^{2} \) |
| 3 | \( 1 - 67.6T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.39e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.13e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.77e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.61e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 6.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.64e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.49e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.40e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.62e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.29e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 7.92e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.69e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.56e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.87e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.09e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.44e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.07e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.68e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.82e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.56e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.64e9T + 7.60e17T^{2} \) |
| 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
−9.525225793014108048826882724570, −8.352438257046033179487546155077, −8.191277734724632826719168542199, −7.13975286596479327613834063880, −5.23754130261193766958922822543, −4.73982111138525176257527948860, −3.75332892337395579520531927838, −2.86896034712441005770682194381, −1.13953791723811443110497613969, 0,
1.13953791723811443110497613969, 2.86896034712441005770682194381, 3.75332892337395579520531927838, 4.73982111138525176257527948860, 5.23754130261193766958922822543, 7.13975286596479327613834063880, 8.191277734724632826719168542199, 8.352438257046033179487546155077, 9.525225793014108048826882724570
