| L(s) = 1 | + 2.35i·2-s − 1.73·3-s − 1.53·4-s − 4.07i·6-s − 6.42i·7-s + 5.79i·8-s + 2.99·9-s + (3.26 + 10.5i)11-s + 2.66·12-s − 7.68i·13-s + 15.1·14-s − 19.7·16-s + 8.15i·17-s + 7.05i·18-s − 30.4i·19-s + ⋯ |
| L(s) = 1 | + 1.17i·2-s − 0.577·3-s − 0.383·4-s − 0.679i·6-s − 0.918i·7-s + 0.724i·8-s + 0.333·9-s + (0.297 + 0.954i)11-s + 0.221·12-s − 0.591i·13-s + 1.08·14-s − 1.23·16-s + 0.479i·17-s + 0.392i·18-s − 1.60i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.638805163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638805163\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-3.26 - 10.5i)T \) |
| good | 2 | \( 1 - 2.35iT - 4T^{2} \) |
| 7 | \( 1 + 6.42iT - 49T^{2} \) |
| 13 | \( 1 + 7.68iT - 169T^{2} \) |
| 17 | \( 1 - 8.15iT - 289T^{2} \) |
| 19 | \( 1 + 30.4iT - 361T^{2} \) |
| 23 | \( 1 - 31.6T + 529T^{2} \) |
| 29 | \( 1 - 6.88iT - 841T^{2} \) |
| 31 | \( 1 - 51.1T + 961T^{2} \) |
| 37 | \( 1 - 19.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 47.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 81.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 30.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 26.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 82.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 75.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 34T + 4.48e3T^{2} \) |
| 71 | \( 1 + 72.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 54.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 24.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 0.923iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 44.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 21.2T + 9.40e3T^{2} \) |
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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
−10.27038490640893499159324389618, −9.359316717113031070682619483839, −8.286351570015974899072830111278, −7.42558284209602923575675824401, −6.82893813748519801150926799723, −6.19449687919121412434771481157, −4.92976645678553326439418435172, −4.50755893619859906816796994460, −2.79529343746089224906864336311, −1.02946983206191163177223232671,
0.75520100956935371851162298349, 1.98161192832399728035283220736, 3.08656092469767234555083715511, 4.06012811316214124312160230884, 5.29401565688610845401670473561, 6.18190055473715647026872207752, 7.01523067728902575705910874309, 8.371672821422296335474021952901, 9.168604798588143406540006828252, 9.994368020967005515000057177957
