| L(s) = 1 | + 19.8i·3-s + (45 − 33.1i)5-s − 59.6i·7-s − 153·9-s − 252·11-s + 119. i·13-s + (660 + 895. i)15-s − 689. i·17-s + 220·19-s + 1.18e3·21-s + 2.43e3i·23-s + (924. − 2.98e3i)25-s + 1.79e3i·27-s + 6.93e3·29-s + 6.75e3·31-s + ⋯ |
| L(s) = 1 | + 1.27i·3-s + (0.804 − 0.593i)5-s − 0.460i·7-s − 0.629·9-s − 0.627·11-s + 0.195i·13-s + (0.757 + 1.02i)15-s − 0.578i·17-s + 0.139·19-s + 0.587·21-s + 0.959i·23-s + (0.295 − 0.955i)25-s + 0.472i·27-s + 1.53·29-s + 1.26·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.451800411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.451800411\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-45 + 33.1i)T \) |
| good | 3 | \( 1 - 19.8iT - 243T^{2} \) |
| 7 | \( 1 + 59.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 252T + 1.61e5T^{2} \) |
| 13 | \( 1 - 119. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 689. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 220T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.39e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 198T + 1.15e8T^{2} \) |
| 43 | \( 1 - 417. iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.05e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 5.82e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.36e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.09e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5iT - 8.58e9T^{2} \) |
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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
−10.54762364542483702189177605132, −9.985222696342128805290826051643, −9.279836798413377180412037199013, −8.322391349587799893879498164950, −6.99452933116971591506472427072, −5.61103525918812690149165152958, −4.87791178937101566643690811348, −3.93362606396282082009758789029, −2.53844537548554532048530836116, −0.875626916871380084664835081593,
0.906969510706608520532566786688, 2.13458075750337432354014031422, 2.92752251636657202497435596053, 4.90140162763634316618110025823, 6.20307045251558901401140122682, 6.60695617428181499303727716187, 7.81633255876696473312750953196, 8.566206397737891162007650538369, 9.936509329006115361329349307374, 10.61885727057638635489099094405
