| L(s) = 1 | + (−2.91 − 15.7i)2-s + (−37.3 − 12.1i)3-s + (−239. + 91.6i)4-s + (614. − 113. i)5-s + (−82.1 + 622. i)6-s + 591. i·7-s + (2.13e3 + 3.49e3i)8-s + (−4.06e3 − 2.94e3i)9-s + (−3.58e3 − 9.33e3i)10-s + (−9.41e3 − 1.29e4i)11-s + (1.00e4 − 522. i)12-s + (−1.64e3 − 1.19e3i)13-s + (9.29e3 − 1.72e3i)14-s + (−2.43e4 − 3.20e3i)15-s + (4.87e4 − 4.38e4i)16-s + (5.08e4 + 1.56e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.182 − 0.983i)2-s + (−0.461 − 0.149i)3-s + (−0.933 + 0.357i)4-s + (0.983 − 0.182i)5-s + (−0.0633 + 0.480i)6-s + 0.246i·7-s + (0.521 + 0.852i)8-s + (−0.618 − 0.449i)9-s + (−0.358 − 0.933i)10-s + (−0.643 − 0.885i)11-s + (0.484 − 0.0251i)12-s + (−0.0576 − 0.0418i)13-s + (0.242 − 0.0448i)14-s + (−0.480 − 0.0632i)15-s + (0.743 − 0.668i)16-s + (0.608 + 1.87i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.19525 - 0.241127i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19525 - 0.241127i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.91 + 15.7i)T \) |
| 5 | \( 1 + (-614. + 113. i)T \) |
| good | 3 | \( 1 + (37.3 + 12.1i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 591. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (9.41e3 + 1.29e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (1.64e3 + 1.19e3i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-5.08e4 - 1.56e5i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (1.39e5 - 4.54e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-8.52e4 - 1.17e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (5.53e4 - 1.70e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (3.40e5 - 1.10e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-9.29e5 - 6.75e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (1.28e5 + 9.37e4i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 3.12e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.82e6 - 1.24e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-8.41e5 + 2.59e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (1.36e6 - 1.87e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (7.75e6 - 5.63e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-7.48e6 + 2.43e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-4.17e7 - 1.35e7i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-3.22e7 + 2.34e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-3.87e7 - 1.26e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (1.92e7 - 6.25e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-2.78e6 + 2.02e6i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (1.77e6 - 5.45e6i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.29529847036956113525537653461, −10.95809937332206484819730856178, −10.35234039593607596624685624224, −9.032788304111798991163669058704, −8.241749691147066167033879703250, −6.11995679462946217035119007726, −5.36141294646564848177752745412, −3.57567848560327987094683355186, −2.20812302816329367599165126548, −0.919261469385458143359052679338,
0.50586553487855133551985533428, 2.45618877664492206126988211143, 4.71664439544165796509551929348, 5.47138554304441486239408477391, 6.65228324683733065783478019082, 7.69765438856451748033516488784, 9.119019784291952737344799560576, 10.01961763813836145784015740401, 10.96552294772709370020439453979, 12.61394419907723117690357690350
