L(s) = 1 | − 17.1·2-s − 137.·3-s − 217.·4-s + 1.24e3·5-s + 2.36e3·6-s + 1.05e4·7-s + 1.25e4·8-s − 717.·9-s − 2.14e4·10-s + 4.94e4·11-s + 3.00e4·12-s + 5.43e4·13-s − 1.80e5·14-s − 1.71e5·15-s − 1.03e5·16-s + 1.23e4·18-s + 7.16e5·19-s − 2.71e5·20-s − 1.45e6·21-s − 8.47e5·22-s − 2.03e5·23-s − 1.72e6·24-s − 3.95e5·25-s − 9.32e5·26-s + 2.80e6·27-s − 2.29e6·28-s − 6.48e6·29-s + ⋯ |
L(s) = 1 | − 0.757·2-s − 0.981·3-s − 0.425·4-s + 0.892·5-s + 0.743·6-s + 1.66·7-s + 1.08·8-s − 0.0364·9-s − 0.676·10-s + 1.01·11-s + 0.417·12-s + 0.527·13-s − 1.25·14-s − 0.876·15-s − 0.393·16-s + 0.0276·18-s + 1.26·19-s − 0.379·20-s − 1.63·21-s − 0.771·22-s − 0.151·23-s − 1.06·24-s − 0.202·25-s − 0.400·26-s + 1.01·27-s − 0.706·28-s − 1.70·29-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 | \( 1 + 17.1T + 512T^{2} \) |
| 3 | \( 1 + 137.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.24e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.05e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.94e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.43e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.16e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.03e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.48e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.52e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.86e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.92e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.47e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 7.91e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.83e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.19e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.69e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.75e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.86e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.69e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.25e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.13e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.81e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.61e9T + 7.60e17T^{2} \) |
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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
−9.695648444223887144472138315967, −8.901976098886306524641683968367, −8.026504944970748350018223734020, −6.90388188706461709300904304691, −5.52420285197011176721112324442, −5.16507730856976155920703215049, −3.86119934409056228421770204830, −1.67996103460406198438937864255, −1.32343872902529739425779824005, 0,
1.32343872902529739425779824005, 1.67996103460406198438937864255, 3.86119934409056228421770204830, 5.16507730856976155920703215049, 5.52420285197011176721112324442, 6.90388188706461709300904304691, 8.026504944970748350018223734020, 8.901976098886306524641683968367, 9.695648444223887144472138315967