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
−10.01238968064144032350441053791, −9.084312880032676944946006094132, −8.217292764198563464910233167195, −7.40877071397896186739530922675, −6.67472557268151412319326831854, −6.26359038267906358870180377897, −4.52669784097054187260737332639, −4.41514800217115166758462968637, −2.98592304974249588401316652127, −0.910243413548004922742034997487,
0.31697348950504972946145365622, 2.47291962920828745434265081908, 3.33445802648622879179822601170, 4.10475548384286496763863680834, 5.33726785387807503550413443762, 6.08831342558755351456032688290, 7.08759672033117912852374628072, 8.230520113856321875825042158543, 8.991477042444933309997086770438, 9.743922257274942166359610803796