| L(s) = 1 | + 22.0·2-s − 192.·3-s − 26.7·4-s − 2.09e3·5-s − 4.23e3·6-s − 1.11e4·7-s − 1.18e4·8-s + 1.72e4·9-s − 4.61e4·10-s + 3.23e4·11-s + 5.14e3·12-s − 5.37e4·13-s − 2.46e5·14-s + 4.02e5·15-s − 2.47e5·16-s + 3.79e5·18-s + 8.34e5·19-s + 5.61e4·20-s + 2.14e6·21-s + 7.12e5·22-s + 8.80e5·23-s + 2.28e6·24-s + 2.43e6·25-s − 1.18e6·26-s + 4.72e5·27-s + 2.99e5·28-s − 6.20e6·29-s + ⋯ |
| L(s) = 1 | + 0.973·2-s − 1.36·3-s − 0.0523·4-s − 1.49·5-s − 1.33·6-s − 1.76·7-s − 1.02·8-s + 0.875·9-s − 1.45·10-s + 0.666·11-s + 0.0716·12-s − 0.522·13-s − 1.71·14-s + 2.05·15-s − 0.944·16-s + 0.851·18-s + 1.46·19-s + 0.0784·20-s + 2.41·21-s + 0.648·22-s + 0.656·23-s + 1.40·24-s + 1.24·25-s − 0.508·26-s + 0.171·27-s + 0.0921·28-s − 1.62·29-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 | \( 1 - 22.0T + 512T^{2} \) |
| 3 | \( 1 + 192.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.09e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.11e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.23e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.37e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 8.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.80e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.20e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.41e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.92e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.42e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.24e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.93e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.32e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.73e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.26e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.10e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.06e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.10e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.11e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.51e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.34e8T + 7.60e17T^{2} \) |
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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
−9.848075794269012241802672264901, −8.988498941883030601107211361712, −7.33111380716735918164930101658, −6.62118571038605185343107118391, −5.66909064891198247979375870562, −4.79717339320078322317786102567, −3.68987696814500736300519283623, −3.20206467232101039422213000351, −0.65763033036753612841971740441, 0,
0.65763033036753612841971740441, 3.20206467232101039422213000351, 3.68987696814500736300519283623, 4.79717339320078322317786102567, 5.66909064891198247979375870562, 6.62118571038605185343107118391, 7.33111380716735918164930101658, 8.988498941883030601107211361712, 9.848075794269012241802672264901
