| L(s) = 1 | + (−173. − 99.9i)5-s + (1.61e3 + 1.77e3i)7-s + (−2.22e4 + 1.28e4i)11-s + 2.64e4·13-s + (−7.42e4 + 4.28e4i)17-s + (−6.94e3 + 1.20e4i)19-s + (6.09e4 + 3.51e4i)23-s + (−1.75e5 − 3.03e5i)25-s + 4.05e5i·29-s + (−1.30e5 − 2.25e5i)31-s + (−1.01e5 − 4.69e5i)35-s + (2.09e5 − 3.62e5i)37-s − 9.27e5i·41-s + 2.87e6·43-s + (1.51e6 + 8.74e5i)47-s + ⋯ |
| L(s) = 1 | + (−0.277 − 0.159i)5-s + (0.671 + 0.740i)7-s + (−1.51 + 0.876i)11-s + 0.927·13-s + (−0.888 + 0.513i)17-s + (−0.0532 + 0.0922i)19-s + (0.217 + 0.125i)23-s + (−0.448 − 0.777i)25-s + 0.573i·29-s + (−0.141 − 0.244i)31-s + (−0.0676 − 0.312i)35-s + (0.111 − 0.193i)37-s − 0.328i·41-s + 0.842·43-s + (0.310 + 0.179i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2654566562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2654566562\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.61e3 - 1.77e3i)T \) |
| good | 5 | \( 1 + (173. + 99.9i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (2.22e4 - 1.28e4i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.64e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (7.42e4 - 4.28e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (6.94e3 - 1.20e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-6.09e4 - 3.51e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 4.05e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (1.30e5 + 2.25e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-2.09e5 + 3.62e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 9.27e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.87e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-1.51e6 - 8.74e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (3.55e6 - 2.04e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-7.54e6 + 4.35e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.68e6 - 8.12e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.71e7 + 2.97e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 9.42e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.48e7 + 2.56e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-3.39e6 + 5.88e6i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 6.35e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (4.29e7 + 2.47e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.06e8T + 7.83e15T^{2} \) |
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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
−10.46715941968552325254778058702, −9.158107759376975399000738875338, −8.287486556265434891717018009806, −7.53502014515082049120745694272, −6.14444629102542680403987437382, −5.13234562624383957945134352250, −4.18960748619579551263262416015, −2.64329424722989642213779243286, −1.68780706756273224123534075810, −0.06105777466896428582974623723,
1.04116379912836607173057381986, 2.51863391760546407710576554548, 3.70772807794894313203755181905, 4.83551352122332162732654100869, 5.89931262951683471128830444036, 7.19924689937586999147641592499, 8.010847520599266581384291787879, 8.844939917998679722417063667249, 10.25668247759936447808108457219, 11.02600350072810518311197879800
