L(s) = 1 | + 4.24i·5-s − 4·13-s + 4.24i·17-s − 12.9·25-s − 4.24i·29-s − 2·37-s + 12.7i·41-s + 7·49-s + 12.7i·53-s + 10·61-s − 16.9i·65-s + 16·73-s − 17.9·85-s − 4.24i·89-s − 8·97-s + ⋯ |
L(s) = 1 | + 1.89i·5-s − 1.10·13-s + 1.02i·17-s − 2.59·25-s − 0.787i·29-s − 0.328·37-s + 1.98i·41-s + 49-s + 1.74i·53-s + 1.28·61-s − 2.10i·65-s + 1.87·73-s − 1.95·85-s − 0.449i·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.510104 + 0.985446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.510104 + 0.985446i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.24iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 12.7iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.94218424541266876694693040738, −10.20922980389346248944772950301, −9.600632917321240083368487812174, −8.155272001130774117625398282381, −7.36335648202768566384320099726, −6.59331539728677042253254300724, −5.77151805299338591681195647618, −4.27094011215157672317506564696, −3.13802009461373358003284695538, −2.21567929648355828327757462051,
0.61016965810068078494371946019, 2.14747593335326535874907222751, 3.88228561683618643959513697057, 5.04012730347137370705953888353, 5.34784401701713482199570853103, 6.93287163823627938508719895635, 7.87570620009373905343372266629, 8.806814450412389348985876620774, 9.364972501136662661314388121361, 10.21556876712294627339832575986