| L(s) = 1 | − 4i·2-s − 13.8i·3-s − 16·4-s − 55.2·6-s + 213. i·7-s + 64i·8-s + 52.2·9-s + 587.·11-s + 220. i·12-s + 169i·13-s + 852.·14-s + 256·16-s + 162. i·17-s − 208. i·18-s + 81.3·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.886i·3-s − 0.5·4-s − 0.626·6-s + 1.64i·7-s + 0.353i·8-s + 0.215·9-s + 1.46·11-s + 0.443i·12-s + 0.277i·13-s + 1.16·14-s + 0.250·16-s + 0.136i·17-s − 0.152i·18-s + 0.0517·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.270069001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.270069001\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 3 | \( 1 + 13.8iT - 243T^{2} \) |
| 7 | \( 1 - 213. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 587.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 162. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 81.3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.94e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.61e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 958.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 169. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.16e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.47e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.63e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.83e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.21e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.65e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 8.22e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.38e4iT - 8.58e9T^{2} \) |
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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
−9.614499714368035077133828830585, −9.261680168323628678120553833585, −8.352541077279392892496237227050, −7.31032991877676392980267919770, −6.29560717895398549048010405453, −5.53423751981796829250013596871, −4.22913414634812011431053649666, −3.09566887412121521806125259749, −1.86574969004136478667268614338, −1.40786269689817442154076074892,
0.27401470461281120399157178104, 1.42617728506997653133270024657, 3.58962519728403182276347230552, 4.07270718919929608224409319568, 4.85068011196085016120713452229, 6.14316714076633762036357705322, 7.07020564204057022138702898781, 7.62404060215958156092268949219, 8.976310899044234277287898423790, 9.492523325373256196950066681003
