L(s) = 1 | − 4.87i·5-s − 11.7·7-s + 9.75i·11-s − 22.6·13-s − 22.1i·17-s + 17.7·19-s + 14.1i·23-s + 1.20·25-s + 20.0i·29-s + 39.7·31-s + 57.5i·35-s − 2.40·37-s + 64.3i·41-s − 3.19·43-s + 41.8i·47-s + ⋯ |
L(s) = 1 | − 0.975i·5-s − 1.68·7-s + 0.886i·11-s − 1.74·13-s − 1.30i·17-s + 0.936·19-s + 0.614i·23-s + 0.0480·25-s + 0.689i·29-s + 1.28·31-s + 1.64i·35-s − 0.0649·37-s + 1.56i·41-s − 0.0742·43-s + 0.890i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9761303077\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9761303077\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4.87iT - 25T^{2} \) |
| 7 | \( 1 + 11.7T + 49T^{2} \) |
| 11 | \( 1 - 9.75iT - 121T^{2} \) |
| 13 | \( 1 + 22.6T + 169T^{2} \) |
| 17 | \( 1 + 22.1iT - 289T^{2} \) |
| 19 | \( 1 - 17.7T + 361T^{2} \) |
| 23 | \( 1 - 14.1iT - 529T^{2} \) |
| 29 | \( 1 - 20.0iT - 841T^{2} \) |
| 31 | \( 1 - 39.7T + 961T^{2} \) |
| 37 | \( 1 + 2.40T + 1.36e3T^{2} \) |
| 41 | \( 1 - 64.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 3.19T + 1.84e3T^{2} \) |
| 47 | \( 1 - 41.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 55.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 111. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 18.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 34.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 87.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 151.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 61.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 72.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.3T + 9.40e3T^{2} \) |
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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
−9.627726135647207689778424232365, −9.287186610289203905331822027353, −7.953269654431238855321134631650, −7.14135658380713817865548926889, −6.50223555695454273667829709796, −5.11811899401695851129424763336, −4.79847217933596630385796533873, −3.35038757233585790783861980824, −2.48327639166800387397039894368, −0.804914850890778932411711137567,
0.40491247981270283916694061408, 2.50647276578129634348582102298, 3.10426346327897135685863972591, 4.05026804549170209153675096445, 5.47678830966665186965807040883, 6.32670867689445088743032657016, 6.88961183828961391015109324318, 7.72322216296373207409974032980, 8.822214306974203960944810899123, 9.732022934131313518966912655041