L(s) = 1 | + 2.42e3i·2-s + (1.00e5 − 1.83e4i)3-s − 3.79e6·4-s − 2.12e7·5-s + (4.46e7 + 2.44e8i)6-s + (5.03e8 − 5.51e8i)7-s − 4.11e9i·8-s + (9.78e9 − 3.70e9i)9-s − 5.16e10i·10-s + 1.35e11i·11-s + (−3.81e11 + 6.97e10i)12-s + 6.90e11i·13-s + (1.33e12 + 1.22e12i)14-s + (−2.14e12 + 3.91e11i)15-s + 2.02e12·16-s + 7.37e12·17-s + ⋯ |
L(s) = 1 | + 1.67i·2-s + (0.983 − 0.179i)3-s − 1.80·4-s − 0.975·5-s + (0.301 + 1.64i)6-s + (0.674 − 0.738i)7-s − 1.35i·8-s + (0.935 − 0.353i)9-s − 1.63i·10-s + 1.57i·11-s + (−1.77 + 0.325i)12-s + 1.39i·13-s + (1.23 + 1.12i)14-s + (−0.959 + 0.175i)15-s + 0.460·16-s + 0.886·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(1.427479899\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.427479899\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.00e5 + 1.83e4i)T \) |
| 7 | \( 1 + (-5.03e8 + 5.51e8i)T \) |
good | 2 | \( 1 - 2.42e3iT - 2.09e6T^{2} \) |
| 5 | \( 1 + 2.12e7T + 4.76e14T^{2} \) |
| 11 | \( 1 - 1.35e11iT - 7.40e21T^{2} \) |
| 13 | \( 1 - 6.90e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 - 7.37e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 6.66e11iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 1.94e13iT - 3.94e28T^{2} \) |
| 29 | \( 1 + 8.36e14iT - 5.13e30T^{2} \) |
| 31 | \( 1 + 6.31e14iT - 2.08e31T^{2} \) |
| 37 | \( 1 + 4.54e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.81e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.33e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.03e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.74e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 - 2.02e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.65e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 + 2.52e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 5.48e18iT - 7.52e38T^{2} \) |
| 73 | \( 1 - 5.62e19iT - 1.34e39T^{2} \) |
| 79 | \( 1 - 8.73e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.61e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.02e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 9.52e20iT - 5.27e41T^{2} \) |
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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
−14.57550761462004226538667316534, −13.63082378449422049875148933759, −11.99232612144501907648364032844, −9.767582311114623968417596197483, −8.423340946271813089077534357173, −7.47547486624080046792585861167, −6.94532452328891964408169123599, −4.70532187583155120155186039680, −3.95118158685905280701992983926, −1.64374673869465544449176848606,
0.31941936432302606765418447905, 1.57767779903293067639693966542, 3.07725311088980968876428203795, 3.49875825595316005545499206335, 5.10859507325252964906544249487, 8.042687474145883253439498483898, 8.679993060383791140793565037452, 10.21616086901211903953619184432, 11.29366295589033051518347860632, 12.32198240078814609687232286057