| L(s) = 1 | + (823. + 1.42e3i)2-s + (−2.10e4 − 1.00e5i)3-s + (−3.09e5 + 5.35e5i)4-s + (−1.17e7 + 2.04e7i)5-s + (1.25e8 − 1.12e8i)6-s + (−3.12e8 − 5.40e8i)7-s + 2.43e9·8-s + (−9.57e9 + 4.20e9i)9-s + (−3.88e10 + 3.81e−6i)10-s + (3.94e10 + 6.83e10i)11-s + (6.01e10 + 1.97e10i)12-s + (9.37e10 − 1.62e11i)13-s + (5.14e11 − 8.90e11i)14-s + (2.29e12 + 7.50e11i)15-s + (2.65e12 + 4.60e12i)16-s + 1.45e13·17-s + ⋯ |
| L(s) = 1 | + (0.568 + 0.985i)2-s + (−0.205 − 0.978i)3-s + (−0.147 + 0.255i)4-s + (−0.539 + 0.934i)5-s + (0.847 − 0.759i)6-s + (−0.417 − 0.723i)7-s + 0.802·8-s + (−0.915 + 0.402i)9-s − 1.22·10-s + (0.458 + 0.794i)11-s + (0.280 + 0.0918i)12-s + (0.188 − 0.326i)13-s + (0.475 − 0.822i)14-s + (1.02 + 0.336i)15-s + (0.603 + 1.04i)16-s + 1.74·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.398171345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.398171345\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.10e4 + 1.00e5i)T \) |
| good | 2 | \( 1 + (-823. - 1.42e3i)T + (-1.04e6 + 1.81e6i)T^{2} \) |
| 5 | \( 1 + (1.17e7 - 2.04e7i)T + (-2.38e14 - 4.12e14i)T^{2} \) |
| 7 | \( 1 + (3.12e8 + 5.40e8i)T + (-2.79e17 + 4.83e17i)T^{2} \) |
| 11 | \( 1 + (-3.94e10 - 6.83e10i)T + (-3.70e21 + 6.40e21i)T^{2} \) |
| 13 | \( 1 + (-9.37e10 + 1.62e11i)T + (-1.23e23 - 2.13e23i)T^{2} \) |
| 17 | \( 1 - 1.45e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.64e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + (1.02e14 - 1.78e14i)T + (-1.97e28 - 3.41e28i)T^{2} \) |
| 29 | \( 1 + (-1.28e15 - 2.21e15i)T + (-2.56e30 + 4.44e30i)T^{2} \) |
| 31 | \( 1 + (-4.86e13 + 8.43e13i)T + (-1.04e31 - 1.80e31i)T^{2} \) |
| 37 | \( 1 + 3.82e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + (-5.42e16 + 9.40e16i)T + (-3.69e33 - 6.39e33i)T^{2} \) |
| 43 | \( 1 + (7.12e15 + 1.23e16i)T + (-1.00e34 + 1.73e34i)T^{2} \) |
| 47 | \( 1 + (-2.07e17 - 3.59e17i)T + (-6.50e34 + 1.12e35i)T^{2} \) |
| 53 | \( 1 - 1.22e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + (4.86e17 - 8.42e17i)T + (-7.70e36 - 1.33e37i)T^{2} \) |
| 61 | \( 1 + (-3.18e18 - 5.51e18i)T + (-1.55e37 + 2.68e37i)T^{2} \) |
| 67 | \( 1 + (-1.17e19 + 2.04e19i)T + (-1.11e38 - 1.92e38i)T^{2} \) |
| 71 | \( 1 + 5.21e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 5.32e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + (2.84e19 + 4.92e19i)T + (-3.54e39 + 6.13e39i)T^{2} \) |
| 83 | \( 1 + (3.60e19 + 6.24e19i)T + (-9.99e39 + 1.73e40i)T^{2} \) |
| 89 | \( 1 + 1.17e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + (-3.36e20 - 5.82e20i)T + (-2.63e41 + 4.56e41i)T^{2} \) |
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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
−16.12986048410524774961134660856, −14.63693646834170614456029518438, −13.75929527486269768368637726841, −12.12301074705081712790842674687, −10.43865049182684670847785458331, −7.52402662698063548756806867042, −7.05901438652469539975402580082, −5.59336915610623289359662189195, −3.46763940264307321441635235501, −1.21797892379899288146171146502,
0.840525275026749482203900874018, 3.02945056749483286432682961788, 4.13777211164346559969645161603, 5.53745530303348657668090034364, 8.447467829012242074296932174165, 9.964671573993279964082256681323, 11.64788056913713739088875410057, 12.28752202512218315047993356203, 14.11508330772660170421178333942, 16.01096140197100964983540878062
