L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.22 − 1.22i)7-s + (−0.707 + 0.707i)8-s + 1.73·14-s − 1.00·16-s + (4.24 + 4.24i)17-s + i·19-s + (−4.24 + 4.24i)23-s + (1.22 + 1.22i)28-s + 10.3·29-s + 7·31-s + (−0.707 − 0.707i)32-s + 6i·34-s + (6.12 − 6.12i)37-s + (−0.707 + 0.707i)38-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.462 − 0.462i)7-s + (−0.250 + 0.250i)8-s + 0.462·14-s − 0.250·16-s + (1.02 + 1.02i)17-s + 0.229i·19-s + (−0.884 + 0.884i)23-s + (0.231 + 0.231i)28-s + 1.92·29-s + 1.25·31-s + (−0.125 − 0.125i)32-s + 1.02i·34-s + (1.00 − 1.00i)37-s + (−0.114 + 0.114i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.331654862\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.331654862\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.22 + 1.22i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 + (4.24 - 4.24i)T - 23iT^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + (-6.12 + 6.12i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (1.22 + 1.22i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (8.48 - 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + (7.34 - 7.34i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (8.57 + 8.57i)T + 73iT^{2} \) |
| 79 | \( 1 + 13iT - 79T^{2} \) |
| 83 | \( 1 + (4.24 - 4.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + (-6.12 + 6.12i)T - 97iT^{2} \) |
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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
−9.896958958574455914133895302645, −8.663621422554465694464098998171, −7.981775820444698373677816682578, −7.40585091238338491122164812463, −6.29203476684140129524831217380, −5.71262008581907666217165721668, −4.60503042239438699530208578477, −3.92495058109933924548194029437, −2.80991711358673535395123777866, −1.31053673216358729449850708030,
0.963472459348031127707773569195, 2.38526298648098267894473480954, 3.15625278893659544408261202372, 4.48269771360144161918265726589, 5.02768023255981129219632551299, 6.05888062853101329011544657526, 6.82308571755243876701753260372, 8.045498817223360315025850260332, 8.560035758843894099158722380767, 9.844569905977226074272131613198