L(s) = 1 | − 1.41·3-s − 4-s + i·5-s + 1.00·9-s + 1.41·12-s + (−0.707 + 0.707i)13-s − 1.41i·15-s + 16-s − 1.41·17-s + 2i·19-s − i·20-s + 1.41·23-s − 25-s − i·31-s − 1.00·36-s − 1.41i·37-s + ⋯ |
L(s) = 1 | − 1.41·3-s − 4-s + i·5-s + 1.00·9-s + 1.41·12-s + (−0.707 + 0.707i)13-s − 1.41i·15-s + 16-s − 1.41·17-s + 2i·19-s − i·20-s + 1.41·23-s − 25-s − i·31-s − 1.00·36-s − 1.41i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03519285380\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03519285380\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
| 31 | \( 1 + iT \) |
good | 2 | \( 1 + T^{2} \) |
| 3 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 - 2iT - T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.839950893161899092385783196996, −9.312501696837907515997710166324, −8.204446814130813598681554664878, −7.32111852431977581249116548381, −6.49860957369408700507686464463, −5.93003974824987697273674769902, −5.01393982702516966652878955767, −4.34156489390107871611728109321, −3.36956349871373830768570155578, −1.85630368309975487466113369917,
0.03629652390274289110268692307, 1.14004336220876583025255489716, 2.97165447919623094562908197246, 4.48652628543504515649648409540, 4.98335495843254955774097140474, 5.18673344248147815268195993086, 6.42485092721915000073327158869, 7.08366216075101416046585234512, 8.321227114128265199325494107183, 8.871502481159826766475230748756