L(s) = 1 | + 966. i·2-s + (5.66e4 − 1.66e4i)3-s + 1.14e5·4-s − 1.69e7i·5-s + (1.60e7 + 5.47e7i)6-s + 2.16e8·7-s + 1.12e9i·8-s + (2.93e9 − 1.88e9i)9-s + 1.63e10·10-s + 1.25e10i·11-s + (6.51e9 − 1.90e9i)12-s − 9.02e10·13-s + 2.09e11i·14-s + (−2.80e11 − 9.57e11i)15-s − 9.65e11·16-s + 5.69e11i·17-s + ⋯ |
L(s) = 1 | + 0.943i·2-s + (0.959 − 0.281i)3-s + 0.109·4-s − 1.73i·5-s + (0.265 + 0.905i)6-s + 0.766·7-s + 1.04i·8-s + (0.841 − 0.540i)9-s + 1.63·10-s + 0.482i·11-s + (0.105 − 0.0308i)12-s − 0.654·13-s + 0.723i·14-s + (−0.487 − 1.66i)15-s − 0.878·16-s + 0.282i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{21}{2})\) |
\(\approx\) |
\(2.55222 + 0.366506i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.55222 + 0.366506i\) |
\(L(11)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.66e4 + 1.66e4i)T \) |
good | 2 | \( 1 - 966. iT - 1.04e6T^{2} \) |
| 5 | \( 1 + 1.69e7iT - 9.53e13T^{2} \) |
| 7 | \( 1 - 2.16e8T + 7.97e16T^{2} \) |
| 11 | \( 1 - 1.25e10iT - 6.72e20T^{2} \) |
| 13 | \( 1 + 9.02e10T + 1.90e22T^{2} \) |
| 17 | \( 1 - 5.69e11iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 4.66e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + 1.67e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 - 5.50e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 + 5.98e14T + 6.71e29T^{2} \) |
| 37 | \( 1 + 6.64e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + 6.99e15iT - 1.80e32T^{2} \) |
| 43 | \( 1 - 1.44e16T + 4.67e32T^{2} \) |
| 47 | \( 1 - 9.26e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 + 1.20e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + 1.03e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 + 3.83e17T + 5.08e35T^{2} \) |
| 67 | \( 1 - 1.36e18T + 3.32e36T^{2} \) |
| 71 | \( 1 + 7.04e17iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 1.09e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + 6.22e18T + 8.96e37T^{2} \) |
| 83 | \( 1 + 1.97e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 - 8.06e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 - 5.28e19T + 5.43e39T^{2} \) |
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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
−20.84714363758827746180983531665, −20.02871330881728904628797055474, −17.45377528884081053627302588523, −15.93476102284430895510705466460, −14.36777956959162849009063378589, −12.44648552627923266768067677672, −8.893942238921179221872692250108, −7.61223187642422132809990586230, −4.94589826862370225574677285229, −1.64550885848364297511903844873,
2.17993105286190126971561791800, 3.40642993138832112737378445879, 7.32756742865891911568767457859, 10.04449514753480083370993853457, 11.36918682935804761884844038010, 14.00144413789072722791623203163, 15.33824220814270624252628896887, 18.45646867332355154489393703598, 19.57876794061439941119529629212, 21.13868261458531292024145354107