L(s) = 1 | + (1 + i)3-s + (1 − i)7-s − i·9-s − 6i·11-s + (1 − i)13-s + (−1 − i)17-s + 4·19-s + 2·21-s + (−5 − 5i)23-s + (4 − 4i)27-s + 8i·29-s + 2i·31-s + (6 − 6i)33-s + (5 + 5i)37-s + 2·39-s + ⋯ |
L(s) = 1 | + (0.577 + 0.577i)3-s + (0.377 − 0.377i)7-s − 0.333i·9-s − 1.80i·11-s + (0.277 − 0.277i)13-s + (−0.242 − 0.242i)17-s + 0.917·19-s + 0.436·21-s + (−1.04 − 1.04i)23-s + (0.769 − 0.769i)27-s + 1.48i·29-s + 0.359i·31-s + (1.04 − 1.04i)33-s + (0.821 + 0.821i)37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82831 - 0.519387i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82831 - 0.519387i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (1 + i)T + 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (5 + 5i)T + 23iT^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + (-5 - 5i)T + 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (3 + 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (7 - 7i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-7 + 7i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (9 - 9i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 - 5i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−10.18656293180407804554371281890, −9.253414012269336771168943351682, −8.525634426571618892253276351088, −7.930608750462495142989773829040, −6.64114012365709273266181391516, −5.78637342642598754756864453424, −4.64257544361796656521375668821, −3.58608957262901790098083189385, −2.90160514683644799411765705493, −0.956690192267222933516784088180,
1.73618760263876091414599865268, 2.39316962120136038256601805581, 3.94881765763319959652114720674, 4.92520628464722690442023488985, 5.98755994956523760813513073468, 7.21360138864090689770640421362, 7.68095093311166738795982055461, 8.493892987237767762568860655333, 9.578548788194541063102053656936, 10.06482760269775380371040344600