| L(s) = 1 | + (−16.2 + 28.1i)2-s + (−30.6 − 53.1i)3-s + (−274. − 474. i)4-s + (1.24e3 + 2.15e3i)5-s + 1.99e3·6-s + (−3.56e3 − 6.18e3i)7-s + 1.18e3·8-s + (7.95e3 − 1.37e4i)9-s − 8.09e4·10-s + 5.32e4·11-s + (−1.68e4 + 2.91e4i)12-s + (−6.29e4 − 1.09e5i)13-s + 2.32e5·14-s + (7.63e4 − 1.32e5i)15-s + (1.21e5 − 2.09e5i)16-s + (1.84e5 − 3.19e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.719 + 1.24i)2-s + (−0.218 − 0.378i)3-s + (−0.535 − 0.927i)4-s + (0.889 + 1.54i)5-s + 0.629·6-s + (−0.561 − 0.972i)7-s + 0.101·8-s + (0.404 − 0.700i)9-s − 2.56·10-s + 1.09·11-s + (−0.234 + 0.405i)12-s + (−0.611 − 1.05i)13-s + 1.61·14-s + (0.389 − 0.674i)15-s + (0.462 − 0.800i)16-s + (0.535 − 0.927i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.17977 + 0.480505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17977 + 0.480505i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (1.04e7 + 4.65e6i)T \) |
| good | 2 | \( 1 + (16.2 - 28.1i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (30.6 + 53.1i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.24e3 - 2.15e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (3.56e3 + 6.18e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 - 5.32e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (6.29e4 + 1.09e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-1.84e5 + 3.19e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.24e5 - 3.88e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + 7.77e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.50e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.16e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (-8.87e6 - 1.53e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 - 2.69e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.14e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-3.22e7 + 5.57e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (2.87e7 - 4.98e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (5.63e7 + 9.76e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.05e8 + 1.82e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-7.56e7 - 1.30e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 7.09e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + (4.93e7 + 8.54e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (2.70e8 - 4.68e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-5.84e8 + 1.01e9i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 4.08e8T + 7.60e17T^{2} \) |
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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
−14.55552778886004071254709686352, −13.91034374083590434145636665545, −12.10679709925864491316059365091, −10.19886022179892180507356603579, −9.644030626424580167188514446462, −7.52794597260013927779783184871, −6.77374783861520422026339811968, −6.02684055727325177748732780208, −3.20191398769508020806101359611, −0.75218700155128046698572849635,
1.14696527712637653978403955384, 2.22459922971188492188147840600, 4.44043404083868483999768933514, 5.95825176886292643121396544734, 8.702378072206276425264807632567, 9.362360865165248182023459990268, 10.20227110246920293522722834183, 11.96791514762109802368075220398, 12.42168583550104985742813778735, 13.82039738337518521750020895025
