L(s) = 1 | + (11.3 − 0.135i)2-s + (26.1 + 26.1i)3-s + (127. − 3.05i)4-s + (−31.6 + 31.6i)5-s + (299. + 292. i)6-s + 444. i·7-s + (1.44e3 − 51.8i)8-s − 820. i·9-s + (−353. + 361. i)10-s + (472. − 472. i)11-s + (3.42e3 + 3.26e3i)12-s + (−2.74e3 − 2.74e3i)13-s + (60.0 + 5.02e3i)14-s − 1.65e3·15-s + (1.63e4 − 782. i)16-s − 2.79e4·17-s + ⋯ |
L(s) = 1 | + (0.999 − 0.0119i)2-s + (0.558 + 0.558i)3-s + (0.999 − 0.0238i)4-s + (−0.113 + 0.113i)5-s + (0.565 + 0.552i)6-s + 0.489i·7-s + (0.999 − 0.0358i)8-s − 0.375i·9-s + (−0.111 + 0.114i)10-s + (0.107 − 0.107i)11-s + (0.572 + 0.545i)12-s + (−0.346 − 0.346i)13-s + (0.00584 + 0.489i)14-s − 0.126·15-s + (0.998 − 0.0477i)16-s − 1.38·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.94730 + 0.659819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.94730 + 0.659819i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-11.3 + 0.135i)T \) |
good | 3 | \( 1 + (-26.1 - 26.1i)T + 2.18e3iT^{2} \) |
| 5 | \( 1 + (31.6 - 31.6i)T - 7.81e4iT^{2} \) |
| 7 | \( 1 - 444. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + (-472. + 472. i)T - 1.94e7iT^{2} \) |
| 13 | \( 1 + (2.74e3 + 2.74e3i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + 2.79e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + (3.57e4 + 3.57e4i)T + 8.93e8iT^{2} \) |
| 23 | \( 1 + 1.89e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + (-1.01e5 - 1.01e5i)T + 1.72e10iT^{2} \) |
| 31 | \( 1 - 2.12e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (3.06e5 - 3.06e5i)T - 9.49e10iT^{2} \) |
| 41 | \( 1 + 1.52e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (3.51e3 - 3.51e3i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 - 5.30e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (1.04e6 - 1.04e6i)T - 1.17e12iT^{2} \) |
| 59 | \( 1 + (1.13e6 - 1.13e6i)T - 2.48e12iT^{2} \) |
| 61 | \( 1 + (-1.95e5 - 1.95e5i)T + 3.14e12iT^{2} \) |
| 67 | \( 1 + (1.52e6 + 1.52e6i)T + 6.06e12iT^{2} \) |
| 71 | \( 1 - 4.95e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 5.65e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 5.88e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + (1.38e6 + 1.38e6i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 - 2.76e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 4.24e6T + 8.07e13T^{2} \) |
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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
−17.37373843845715330000267901996, −15.52819399208359090047372671185, −15.11611633756207793875694379194, −13.64217788465693675144860310416, −12.22687300965609890975371229743, −10.70753419401637342094324101626, −8.821070917695681058409393790262, −6.61550578772967177443281877804, −4.55593171561648294461514532515, −2.79105941828633806011252030319,
2.16040250418056141149621549132, 4.36039301134535876329581533167, 6.62207202059194736113272432704, 8.118781358288492901138633453367, 10.58970475004342795600416874417, 12.23774517764295380004140998142, 13.46016340401633083384563133857, 14.35150579216875038686281339339, 15.81267189764415568439969737625, 17.19173199218317282510457440021