L(s) = 1 | + (0.923 + 0.923i)3-s + (0.915 − 2.48i)7-s − 1.29i·9-s + 1.21·11-s + (0.996 + 0.996i)13-s + (0.567 − 0.567i)17-s + 0.103·19-s + (3.13 − 1.44i)21-s + (−1.43 + 1.43i)23-s + (3.96 − 3.96i)27-s − 4.97i·29-s − 4.35i·31-s + (1.12 + 1.12i)33-s + (2.54 + 2.54i)37-s + 1.83i·39-s + ⋯ |
L(s) = 1 | + (0.532 + 0.532i)3-s + (0.345 − 0.938i)7-s − 0.431i·9-s + 0.366·11-s + (0.276 + 0.276i)13-s + (0.137 − 0.137i)17-s + 0.0237·19-s + (0.684 − 0.315i)21-s + (−0.298 + 0.298i)23-s + (0.763 − 0.763i)27-s − 0.924i·29-s − 0.781i·31-s + (0.195 + 0.195i)33-s + (0.417 + 0.417i)37-s + 0.294i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.122570355\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.122570355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.915 + 2.48i)T \) |
good | 3 | \( 1 + (-0.923 - 0.923i)T + 3iT^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + (-0.996 - 0.996i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.567 + 0.567i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.103T + 19T^{2} \) |
| 23 | \( 1 + (1.43 - 1.43i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.97iT - 29T^{2} \) |
| 31 | \( 1 + 4.35iT - 31T^{2} \) |
| 37 | \( 1 + (-2.54 - 2.54i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.06iT - 41T^{2} \) |
| 43 | \( 1 + (3.11 - 3.11i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.23 + 7.23i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.882 - 0.882i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.28T + 59T^{2} \) |
| 61 | \( 1 + 10.0iT - 61T^{2} \) |
| 67 | \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.329T + 71T^{2} \) |
| 73 | \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \) |
| 79 | \( 1 - 13.6iT - 79T^{2} \) |
| 83 | \( 1 + (-4.61 - 4.61i)T + 83iT^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + (2.40 - 2.40i)T - 97iT^{2} \) |
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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
−9.699559852258327439529137020515, −8.689180554531742569249947560531, −8.029594922439226675404286764043, −7.08632793210726222542264384405, −6.32458832568693589615332130058, −5.19347556820502998202985147780, −4.00274905916682925273207175587, −3.78893221725632158354809792327, −2.37131987550211202828234901402, −0.897586441396017728825779646072,
1.42048913915827499452254530056, 2.39803532243645805832253958542, 3.32396661360207236250371889129, 4.63853118528633590184488755515, 5.49123123048758643075087310942, 6.38303649473501196600324545382, 7.32800181321314910295798552675, 8.131683229277186096891160789003, 8.675835660988076945556957620517, 9.397374112851073309938776637309