L(s) = 1 | + (−0.438 − 0.253i)3-s + (4.59 − 2.65i)5-s + (−5.27 + 4.59i)7-s + (−4.37 − 7.57i)9-s + (−8.54 + 14.8i)11-s + 21.4i·13-s − 2.68·15-s + (−20.7 − 12.0i)17-s + (10.5 − 6.11i)19-s + (3.48 − 0.680i)21-s + (20.1 + 34.8i)23-s + (1.55 − 2.69i)25-s + 8.99i·27-s + 26.0·29-s + (−21.8 − 12.6i)31-s + ⋯ |
L(s) = 1 | + (−0.146 − 0.0844i)3-s + (0.918 − 0.530i)5-s + (−0.753 + 0.656i)7-s + (−0.485 − 0.841i)9-s + (−0.776 + 1.34i)11-s + 1.65i·13-s − 0.179·15-s + (−1.22 − 0.706i)17-s + (0.557 − 0.321i)19-s + (0.165 − 0.0324i)21-s + (0.875 + 1.51i)23-s + (0.0623 − 0.107i)25-s + 0.333i·27-s + 0.899·29-s + (−0.705 − 0.407i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9637992031\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9637992031\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (5.27 - 4.59i)T \) |
good | 3 | \( 1 + (0.438 + 0.253i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-4.59 + 2.65i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (8.54 - 14.8i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 21.4iT - 169T^{2} \) |
| 17 | \( 1 + (20.7 + 12.0i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-10.5 + 6.11i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-20.1 - 34.8i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 26.0T + 841T^{2} \) |
| 31 | \( 1 + (21.8 + 12.6i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-6.48 - 11.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 33.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 29.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (48.2 - 27.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (4.36 - 7.55i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-43.2 - 24.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (3.40 - 1.96i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (52.9 - 91.6i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 35.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-40.3 - 23.3i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (43.4 + 75.3i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 64.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-37.2 + 21.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 28.7iT - 9.40e3T^{2} \) |
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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
−11.45080590722750276112104994922, −9.844959807915324626263533744222, −9.374228298685148223483069126368, −8.892103670473456411718792490731, −7.19185387612281950135763024824, −6.51743576714067810265158824692, −5.46415408791127061500447050205, −4.57516571903674729903845211599, −2.89900031180577925372540290922, −1.73567636149457240657806690980,
0.38334458258137809137517049871, 2.51666639328654133370944388851, 3.32711250226081262572379349123, 5.05774005678415825629058893575, 5.90915742476095043200210155285, 6.67759813811856150088061275579, 7.984700928989802508584515768399, 8.697526634621363704394325418678, 10.11183580501452752447953811154, 10.62095706624727271287843421962