L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.258 + 0.965i)5-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s − i·10-s + (0.965 + 0.258i)11-s + (−0.366 − 1.36i)13-s + (−0.965 + 0.258i)14-s + (0.500 + 0.866i)16-s − 1.41i·17-s + (−0.258 + 0.965i)20-s + (−0.866 − 0.499i)22-s + 1.41i·26-s + 28-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.258 + 0.965i)5-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s − i·10-s + (0.965 + 0.258i)11-s + (−0.366 − 1.36i)13-s + (−0.965 + 0.258i)14-s + (0.500 + 0.866i)16-s − 1.41i·17-s + (−0.258 + 0.965i)20-s + (−0.866 − 0.499i)22-s + 1.41i·26-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8434226915\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8434226915\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.901878093510681805461718728177, −9.131888014654108073828470627271, −8.178557610042001960663756705635, −7.34442319017368317816503479465, −6.95807306668312226574116559695, −5.87771313823669748689660824964, −4.66991467066191605550184701764, −3.34176612729955676184232078924, −2.55175205032466610677713587452, −1.20840452067686948604551744019,
1.43828555869461774644098270124, 2.07608058654106395792654835906, 3.93253946726014538604070551864, 4.96604598612155924468615403341, 5.86713011692606570304052165716, 6.62688775960808152726352764288, 7.64740416039618953946815766017, 8.541871903472886613452739409595, 8.957138947951810110100819612762, 9.528155937810069248643874782123