L(s) = 1 | − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−2.07 + 0.837i)5-s + (0.707 + 0.707i)6-s + (−2.93 + 2.93i)7-s + i·8-s − 1.00i·9-s + (0.837 + 2.07i)10-s − 1.12i·11-s + (0.707 − 0.707i)12-s − 2.10i·13-s + (2.93 + 2.93i)14-s + (0.874 − 2.05i)15-s + 16-s − 3.81·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−0.927 + 0.374i)5-s + (0.288 + 0.288i)6-s + (−1.11 + 1.11i)7-s + 0.353i·8-s − 0.333i·9-s + (0.264 + 0.655i)10-s − 0.340i·11-s + (0.204 − 0.204i)12-s − 0.583i·13-s + (0.785 + 0.785i)14-s + (0.225 − 0.531i)15-s + 0.250·16-s − 0.926·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0180 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0180 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5013745646\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5013745646\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.07 - 0.837i)T \) |
| 37 | \( 1 + (-5.91 + 1.41i)T \) |
good | 7 | \( 1 + (2.93 - 2.93i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.12iT - 11T^{2} \) |
| 13 | \( 1 + 2.10iT - 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 + (0.0424 - 0.0424i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.44iT - 23T^{2} \) |
| 29 | \( 1 + (-1.00 - 1.00i)T + 29iT^{2} \) |
| 31 | \( 1 + (-5.55 + 5.55i)T - 31iT^{2} \) |
| 41 | \( 1 + 7.31iT - 41T^{2} \) |
| 43 | \( 1 + 6.59iT - 43T^{2} \) |
| 47 | \( 1 + (-3.52 + 3.52i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.43 + 6.43i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.53 + 1.53i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.13 + 3.13i)T - 61iT^{2} \) |
| 67 | \( 1 + (-2.59 - 2.59i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.10T + 71T^{2} \) |
| 73 | \( 1 + (6.95 - 6.95i)T - 73iT^{2} \) |
| 79 | \( 1 + (8.44 - 8.44i)T - 79iT^{2} \) |
| 83 | \( 1 + (2.80 + 2.80i)T + 83iT^{2} \) |
| 89 | \( 1 + (8.09 + 8.09i)T + 89iT^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.743922257274942166359610803796, −8.991477042444933309997086770438, −8.230520113856321875825042158543, −7.08759672033117912852374628072, −6.08831342558755351456032688290, −5.33726785387807503550413443762, −4.10475548384286496763863680834, −3.33445802648622879179822601170, −2.47291962920828745434265081908, −0.31697348950504972946145365622,
0.910243413548004922742034997487, 2.98592304974249588401316652127, 4.41514800217115166758462968637, 4.52669784097054187260737332639, 6.26359038267906358870180377897, 6.67472557268151412319326831854, 7.40877071397896186739530922675, 8.217292764198563464910233167195, 9.084312880032676944946006094132, 10.01238968064144032350441053791