L(s) = 1 | + (−0.531 − 1.31i)2-s + (−1.43 + 1.39i)4-s − i·5-s + 1.38i·7-s + (2.58 + 1.13i)8-s + (−1.31 + 0.531i)10-s − 6.29·11-s + 5.01·13-s + (1.81 − 0.734i)14-s + (0.117 − 3.99i)16-s − 1.58i·17-s + 0.313i·19-s + (1.39 + 1.43i)20-s + (3.34 + 8.24i)22-s + 0.478·23-s + ⋯ |
L(s) = 1 | + (−0.375 − 0.926i)2-s + (−0.717 + 0.696i)4-s − 0.447i·5-s + 0.522i·7-s + (0.915 + 0.402i)8-s + (−0.414 + 0.168i)10-s − 1.89·11-s + 1.38·13-s + (0.484 − 0.196i)14-s + (0.0294 − 0.999i)16-s − 0.384i·17-s + 0.0720i·19-s + (0.311 + 0.320i)20-s + (0.713 + 1.75i)22-s + 0.0997·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147761205\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147761205\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.531 + 1.31i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - 1.38iT - 7T^{2} \) |
| 11 | \( 1 + 6.29T + 11T^{2} \) |
| 13 | \( 1 - 5.01T + 13T^{2} \) |
| 17 | \( 1 + 1.58iT - 17T^{2} \) |
| 19 | \( 1 - 0.313iT - 19T^{2} \) |
| 23 | \( 1 - 0.478T + 23T^{2} \) |
| 29 | \( 1 - 1.71iT - 29T^{2} \) |
| 31 | \( 1 - 9.47iT - 31T^{2} \) |
| 37 | \( 1 - 8.55T + 37T^{2} \) |
| 41 | \( 1 + 7.54iT - 41T^{2} \) |
| 43 | \( 1 + 6.02iT - 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 8.86iT - 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 5.02T + 61T^{2} \) |
| 67 | \( 1 - 3.38iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 + 2.85iT - 79T^{2} \) |
| 83 | \( 1 + 0.983T + 83T^{2} \) |
| 89 | \( 1 - 2.64iT - 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.151622319698450175371167486745, −8.693551932386144543423238156092, −8.006744549578935019293837960800, −7.17535676826844751970003003453, −5.70275607421830925349310940980, −5.16314315479728978694779356714, −4.08813990623630489760134350362, −3.03951455370895314720370992875, −2.20691721345323312309518986167, −0.845559154740338550824979628135,
0.74674051322684321982708936655, 2.39104735756113271588783592423, 3.73501593884644679723994930655, 4.63595623081406402692197738555, 5.72740942364919741468494954345, 6.19574351045004970095854593433, 7.24441350999875954124312546980, 7.939425989535259847638827812032, 8.348820060647518118777902842451, 9.509114525619463854304833744459