L(s) = 1 | + 3.08i·2-s − 1.53·4-s + (−5.83 + 9.53i)5-s + 31.3i·7-s + 19.9i·8-s + (−29.4 − 18.0i)10-s − 19.1·11-s − 20.9i·13-s − 96.6·14-s − 73.9·16-s + 6.19i·17-s + 96.6·19-s + (8.94 − 14.6i)20-s − 59.2i·22-s + 162. i·23-s + ⋯ |
L(s) = 1 | + 1.09i·2-s − 0.191·4-s + (−0.522 + 0.852i)5-s + 1.69i·7-s + 0.882i·8-s + (−0.930 − 0.569i)10-s − 0.526·11-s − 0.447i·13-s − 1.84·14-s − 1.15·16-s + 0.0883i·17-s + 1.16·19-s + (0.0999 − 0.163i)20-s − 0.574i·22-s + 1.47i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.367666274\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.367666274\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (5.83 - 9.53i)T \) |
good | 2 | \( 1 - 3.08iT - 8T^{2} \) |
| 7 | \( 1 - 31.3iT - 343T^{2} \) |
| 11 | \( 1 + 19.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 6.19iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 96.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 162. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 7.64T + 2.43e4T^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 155. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 192. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 318. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 277. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 429.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 89.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 583. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 259. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 124.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 36.7iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 333.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3iT - 9.12e5T^{2} \) |
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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
−11.73952509126665834060376551684, −10.54709815765505532423113283784, −9.389585267744648505243833171978, −8.336162472539263917493984273759, −7.70358809745135241593402372049, −6.76827223768931724282133351549, −5.74403668773748917870507784261, −5.19317539668540732863389724180, −3.26788200183086249044650671471, −2.29091835637323874019171129800,
0.47389799149398504740708560093, 1.33906801171567710063387281682, 3.02083594635107758888754971675, 4.10590615807670455317372551911, 4.80855055922249534574358489618, 6.62047416551655501074017380918, 7.48177916375458848719527316796, 8.440038876269831237581602116215, 9.759250029084080314064942322460, 10.27011249904469684809688523750