L(s) = 1 | + (−8.14e4 − 6.18e4i)3-s + 1.06e7i·5-s + 3.89e8i·7-s + (2.80e9 + 1.00e10i)9-s + 1.23e11·11-s − 7.26e11·13-s + (6.55e11 − 8.63e11i)15-s + 1.04e13i·17-s + 4.84e13i·19-s + (2.40e13 − 3.17e13i)21-s + 2.08e13·23-s + 3.64e14·25-s + (3.95e14 − 9.94e14i)27-s + 1.94e15i·29-s + 4.62e15i·31-s + ⋯ |
L(s) = 1 | + (−0.796 − 0.604i)3-s + 0.485i·5-s + 0.521i·7-s + (0.267 + 0.963i)9-s + 1.43·11-s − 1.46·13-s + (0.293 − 0.386i)15-s + 1.25i·17-s + 1.81i·19-s + (0.315 − 0.414i)21-s + 0.104·23-s + 0.764·25-s + (0.369 − 0.929i)27-s + 0.857i·29-s + 1.01i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(1.102894301\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.102894301\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.14e4 + 6.18e4i)T \) |
good | 5 | \( 1 - 1.06e7iT - 4.76e14T^{2} \) |
| 7 | \( 1 - 3.89e8iT - 5.58e17T^{2} \) |
| 11 | \( 1 - 1.23e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.26e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.04e13iT - 6.90e25T^{2} \) |
| 19 | \( 1 - 4.84e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 2.08e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.94e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 - 4.62e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 - 3.28e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.13e17iT - 7.38e33T^{2} \) |
| 43 | \( 1 + 7.69e16iT - 2.00e34T^{2} \) |
| 47 | \( 1 + 5.40e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.83e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 - 5.25e17T + 1.54e37T^{2} \) |
| 61 | \( 1 + 4.40e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.76e17iT - 2.22e38T^{2} \) |
| 71 | \( 1 - 2.91e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.60e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 9.16e19iT - 7.08e39T^{2} \) |
| 83 | \( 1 + 9.06e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 5.08e20iT - 8.65e40T^{2} \) |
| 97 | \( 1 + 2.06e20T + 5.27e41T^{2} \) |
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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
−12.18219852129212414778358161865, −10.90717089087164858782210517482, −9.853911420738051535460004285009, −8.368794434268282418158635515906, −7.07596862612543396008294767372, −6.26555526520540079606473295882, −5.15944613275024277056544999664, −3.71308263914330819611338626756, −2.14548395895626259669991739428, −1.24140864505048067102793677471,
0.30509053525290217559541911003, 0.988623534545447322839015946955, 2.80142847574154261288810604652, 4.40564940360503324830275656200, 4.84410677471686062322275678047, 6.38924384891045359435608394923, 7.33689274386024922814392152006, 9.228780078269758883764161062571, 9.696921952902969385232850684890, 11.24524163533971854992037455099