L(s) = 1 | + (−1.36e3 − 478. i)2-s + (1.63e6 + 1.30e6i)4-s − 1.38e7·5-s + 1.09e9i·7-s + (−1.61e9 − 2.57e9i)8-s + (1.88e10 + 6.60e9i)10-s + 4.42e10i·11-s − 9.88e10i·13-s + (5.22e11 − 1.49e12i)14-s + (9.78e11 + 4.28e12i)16-s − 4.30e12i·17-s − 6.19e12·19-s + (−2.26e13 − 1.80e13i)20-s + (2.11e13 − 6.05e13i)22-s + 1.44e14·23-s + ⋯ |
L(s) = 1 | + (−0.943 − 0.330i)2-s + (0.781 + 0.623i)4-s − 0.632·5-s + 1.46i·7-s + (−0.532 − 0.846i)8-s + (0.597 + 0.208i)10-s + 0.514i·11-s − 0.198i·13-s + (0.482 − 1.37i)14-s + (0.222 + 0.974i)16-s − 0.518i·17-s − 0.231·19-s + (−0.494 − 0.394i)20-s + (0.169 − 0.485i)22-s + 0.727·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(0.02056406791\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02056406791\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36e3 + 478. i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.38e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 1.09e9iT - 5.58e17T^{2} \) |
| 11 | \( 1 - 4.42e10iT - 7.40e21T^{2} \) |
| 13 | \( 1 + 9.88e10iT - 2.47e23T^{2} \) |
| 17 | \( 1 + 4.30e12iT - 6.90e25T^{2} \) |
| 19 | \( 1 + 6.19e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.44e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.75e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 3.72e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 - 3.46e16iT - 8.55e32T^{2} \) |
| 41 | \( 1 - 1.63e16iT - 7.38e33T^{2} \) |
| 43 | \( 1 + 1.19e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.37e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 3.01e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 1.83e18iT - 1.54e37T^{2} \) |
| 61 | \( 1 + 1.78e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 + 1.98e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.06e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 8.70e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 3.94e19iT - 7.08e39T^{2} \) |
| 83 | \( 1 - 5.89e19iT - 1.99e40T^{2} \) |
| 89 | \( 1 - 3.06e20iT - 8.65e40T^{2} \) |
| 97 | \( 1 - 8.97e20T + 5.27e41T^{2} \) |
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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
−10.26403454062706405550453944943, −9.192245091056178340081448671465, −8.431531706700713644663151347628, −7.41595308184708066923703852518, −6.26164745358750877098285918276, −4.89261115620107809617030952227, −3.34257488040877385429899859169, −2.46397985181050862727732812147, −1.39458874428113589378849582360, −0.00746980967300814516617844286,
0.68076613142950523384761081668, 1.75909580150700093699538524152, 3.32290696029129225600824502739, 4.39667926460702973533418718652, 5.93528957776740292308350167251, 7.08192430928741164890066569922, 7.75532249524154713911521915005, 8.777456231070154193547203639811, 10.02160760526458356722103318806, 10.86132388847769170892640132869