| L(s) = 1 | + 11.3i·2-s − 128.·4-s − 654. i·5-s + 1.29e3·7-s − 1.44e3i·8-s + 7.40e3·10-s − 4.40e3i·11-s − 246.·13-s + 1.47e4i·14-s + 1.63e4·16-s − 8.90e4i·17-s + 1.45e5·19-s + 8.37e4i·20-s + 4.98e4·22-s + 3.55e5i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.04i·5-s + 0.541·7-s − 0.353i·8-s + 0.740·10-s − 0.301i·11-s − 0.00861·13-s + 0.382i·14-s + 0.250·16-s − 1.06i·17-s + 1.11·19-s + 0.523i·20-s + 0.212·22-s + 1.27i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.394754426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394754426\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 11.3iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 654. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.29e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 4.40e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 246.T + 8.15e8T^{2} \) |
| 17 | \( 1 + 8.90e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.45e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 3.55e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 6.35e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 9.72e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 4.08e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.59e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.83e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.25e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.23e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 3.30e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.51e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.76e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.66e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.28e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.49e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 9.14e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.99e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.07e8T + 7.83e15T^{2} \) |
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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
−11.26429441904918535027431173198, −9.718698351105912329435091069472, −8.955349121603095698394274268134, −7.977987845344614154018112228320, −7.04006748655955136255151245436, −5.44408177573993765723074110979, −4.96622159822731135247967564470, −3.49317481692258953464687564948, −1.53092308033709128563808825005, −0.35595863417613802871883126837,
1.35074560142666889127378069781, 2.55611855536053374170899808270, 3.65585765890095961972491350134, 4.94418041718299160267072052098, 6.33811071964865885038713283558, 7.51920531296853240942023673201, 8.614552636997785830898385132934, 9.904885100501877627620946674657, 10.67579225061971742916399660718, 11.44199592044889091871411513110
