L(s) = 1 | − 14i·3-s + (55 + 10i)5-s + 158i·7-s + 47·9-s + 148·11-s + 684i·13-s + (140 − 770i)15-s − 2.04e3i·17-s + 2.22e3·19-s + 2.21e3·21-s + 1.24e3i·23-s + (2.92e3 + 1.10e3i)25-s − 4.06e3i·27-s + 270·29-s + 2.04e3·31-s + ⋯ |
L(s) = 1 | − 0.898i·3-s + (0.983 + 0.178i)5-s + 1.21i·7-s + 0.193·9-s + 0.368·11-s + 1.12i·13-s + (0.160 − 0.883i)15-s − 1.71i·17-s + 1.41·19-s + 1.09·21-s + 0.491i·23-s + (0.936 + 0.352i)25-s − 1.07i·27-s + 0.0596·29-s + 0.382·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.24836 - 0.202734i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.24836 - 0.202734i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-55 - 10i)T \) |
good | 3 | \( 1 + 14iT - 243T^{2} \) |
| 7 | \( 1 - 158iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 148T + 1.61e5T^{2} \) |
| 13 | \( 1 - 684iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.04e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.22e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.24e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 270T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.37e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 2.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.29e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.06e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.96e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.20e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.01e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 8.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.72e4iT - 8.58e9T^{2} \) |
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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
−13.58248784421289701729257868103, −12.19671509682654663964408871948, −11.59846747118026925394831266567, −9.723601451032213785933466091870, −9.051773555324886597155218867365, −7.34926583302124053202932681555, −6.37389826249377757064204605115, −5.12958350681591005540140504081, −2.66908325633065254032997145101, −1.43484698478187625651289244074,
1.22449497840952489148991991128, 3.50316828688008904089271131000, 4.79017797052126808819223439258, 6.16985539287195084585772662105, 7.70784665236488771735719467181, 9.246543034284442442540938365465, 10.27991311618157684912840832220, 10.67475989277613844932529892691, 12.56767908793262022207124289458, 13.50659965957190715628203683561