| L(s) = 1 | + (−1.70 + 0.292i)3-s + (3 + 3i)7-s + (2.82 − i)9-s + 1.41i·11-s + (4.24 − 4.24i)17-s + 4i·19-s + (−5.99 − 4.24i)21-s + (−2.82 − 2.82i)23-s + (−4.53 + 2.53i)27-s − 1.41·29-s + 2·31-s + (−0.414 − 2.41i)33-s + (2 + 2i)37-s + 5.65i·41-s + (−2 + 2i)43-s + ⋯ |
| L(s) = 1 | + (−0.985 + 0.169i)3-s + (1.13 + 1.13i)7-s + (0.942 − 0.333i)9-s + 0.426i·11-s + (1.02 − 1.02i)17-s + 0.917i·19-s + (−1.30 − 0.925i)21-s + (−0.589 − 0.589i)23-s + (−0.872 + 0.487i)27-s − 0.262·29-s + 0.359·31-s + (−0.0721 − 0.420i)33-s + (0.328 + 0.328i)37-s + 0.883i·41-s + (−0.304 + 0.304i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321801253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321801253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.292i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (-2 - 2i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (2 - 2i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.65 + 5.65i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.48 - 8.48i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-4 - 4i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (3 - 3i)T - 73iT^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 + (-2.82 - 2.82i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + (-13 - 13i)T + 97iT^{2} \) |
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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
−9.984150424755847331133915260769, −9.209964019123954904201675630923, −8.168645109228681789967769975155, −7.50195944340190714890507442568, −6.37803349088814428252209582392, −5.52913630451042555307627608728, −5.01433375717286440336004035256, −4.04325814336621241718168657066, −2.50262860453763827825448674382, −1.27530577881669298596963275819,
0.75986099135028635278330319516, 1.81345319030841354769446526460, 3.67316399496878847773510265373, 4.48593962508274016157016990480, 5.37471974882660197280746497393, 6.15222279721812443143147545511, 7.25606774870423548395693883518, 7.70808307674816658979428510382, 8.641101071655318229492067861987, 9.917892691944210939242223786562
