| L(s) = 1 | + 1.41·5-s + 10i·13-s + 9.89i·17-s − 23·25-s − 1.41·29-s + 24i·37-s + 43.8i·41-s − 49·49-s + 103.·53-s + 120i·61-s + 14.1i·65-s − 96·73-s + 14.0i·85-s + 57.9i·89-s − 144·97-s + ⋯ |
| L(s) = 1 | + 0.282·5-s + 0.769i·13-s + 0.582i·17-s − 0.920·25-s − 0.0487·29-s + 0.648i·37-s + 1.06i·41-s − 0.999·49-s + 1.94·53-s + 1.96i·61-s + 0.217i·65-s − 1.31·73-s + 0.164i·85-s + 0.651i·89-s − 1.48·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.394893912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394893912\) |
| \(L(2)\) |
|
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 | \( 1 - 1.41T + 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 - 9.89iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 1.41T + 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 - 24iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 43.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 103.T + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 120iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 57.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 144T + 9.40e3T^{2} \) |
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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
−9.859343978332983778916878888891, −9.030679050008556790515759595119, −8.256856270536130948026759063563, −7.33509018174153780429897446463, −6.44132915302646410752280583769, −5.69679986082384336136700266461, −4.61146326919192269816293046015, −3.73110585597308865210701653239, −2.47221788160497823330499447943, −1.36438076980014877138084177606,
0.41873058858541769324935210586, 1.91588778574575743945702042982, 3.03732579121269054906311726144, 4.08984967877816730073850536097, 5.22926650391183677909339365195, 5.88362042695444301390147581064, 6.93905497136170600721351120531, 7.72223838126363760058529039758, 8.571601204501861372931331860158, 9.453125482301008572154930321450
