L(s) = 1 | + (−3.10e4 + 9.74e4i)3-s + 2.82e7i·5-s − 1.02e8i·7-s + (−8.53e9 − 6.04e9i)9-s − 1.32e11·11-s − 9.58e11·13-s + (−2.75e12 − 8.76e11i)15-s + 2.11e12i·17-s + 5.68e12i·19-s + (1.00e13 + 3.19e12i)21-s + 9.10e13·23-s − 3.22e14·25-s + (8.53e14 − 6.44e14i)27-s − 3.40e15i·29-s + 2.15e15i·31-s + ⋯ |
L(s) = 1 | + (−0.303 + 0.952i)3-s + 1.29i·5-s − 0.137i·7-s + (−0.816 − 0.577i)9-s − 1.54·11-s − 1.92·13-s + (−1.23 − 0.392i)15-s + 0.254i·17-s + 0.212i·19-s + (0.131 + 0.0417i)21-s + 0.458·23-s − 0.676·25-s + (0.797 − 0.602i)27-s − 1.50i·29-s + 0.471i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(0.3414751411\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3414751411\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.10e4 - 9.74e4i)T \) |
good | 5 | \( 1 - 2.82e7iT - 4.76e14T^{2} \) |
| 7 | \( 1 + 1.02e8iT - 5.58e17T^{2} \) |
| 11 | \( 1 + 1.32e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 9.58e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.11e12iT - 6.90e25T^{2} \) |
| 19 | \( 1 - 5.68e12iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 9.10e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.40e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 - 2.15e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 + 2.84e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 2.99e16iT - 7.38e33T^{2} \) |
| 43 | \( 1 - 6.56e16iT - 2.00e34T^{2} \) |
| 47 | \( 1 + 6.12e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.81e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 4.67e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 6.14e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.06e19iT - 2.22e38T^{2} \) |
| 71 | \( 1 - 2.58e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.97e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.51e20iT - 7.08e39T^{2} \) |
| 83 | \( 1 + 1.89e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.33e20iT - 8.65e40T^{2} \) |
| 97 | \( 1 - 4.64e20T + 5.27e41T^{2} \) |
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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
−11.18962493253965254845714209536, −10.33071817838636950986341660664, −9.740136917548602397884775388603, −7.974491598388508357306268940097, −6.90163806351802854592593310629, −5.54938984044689641504611675983, −4.55922748555198817517832428947, −3.09605598338446094130087069356, −2.43919066417968409028955613593, −0.13149689927439162034242627246,
0.49323847928502574950054888946, 1.80605369090665479272473843569, 2.79071803711041124939423367671, 5.08728722175228938839336596820, 5.17108697256379303883815560284, 7.02450016017151463396206861701, 7.913539395382727465411355965455, 8.967869989869105354902179628242, 10.33643739369276472792364681109, 11.79489607715848421170817800435