L(s) = 1 | + (−0.541 − 1.30i)2-s + (−1.64 − 0.541i)3-s + (−1.41 + 1.41i)4-s + (0.183 + 2.44i)6-s − 3.29i·7-s + (2.61 + 1.08i)8-s + (2.41 + 1.78i)9-s − 2.51i·11-s + (3.09 − 1.56i)12-s − 4.65i·13-s + (−4.29 + 1.78i)14-s − 4i·16-s + 3.69i·17-s + (1.02 − 4.11i)18-s − 0.828·19-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.949 − 0.312i)3-s + (−0.707 + 0.707i)4-s + (0.0748 + 0.997i)6-s − 1.24i·7-s + (0.923 + 0.382i)8-s + (0.804 + 0.593i)9-s − 0.759i·11-s + (0.892 − 0.450i)12-s − 1.29i·13-s + (−1.14 + 0.475i)14-s − i·16-s + 0.896i·17-s + (0.240 − 0.970i)18-s − 0.190·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.142594 + 0.384374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.142594 + 0.384374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.541 + 1.30i)T \) |
| 3 | \( 1 + (1.64 + 0.541i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.29iT - 7T^{2} \) |
| 11 | \( 1 + 2.51iT - 11T^{2} \) |
| 13 | \( 1 + 4.65iT - 13T^{2} \) |
| 17 | \( 1 - 3.69iT - 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 1.92iT - 37T^{2} \) |
| 41 | \( 1 - 8.59iT - 41T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 + 2.61T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 + 2.51iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 3.29T + 67T^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 + 16.4iT - 79T^{2} \) |
| 83 | \( 1 - 9.37iT - 83T^{2} \) |
| 89 | \( 1 - 5.03iT - 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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
−10.42328896740673806091765871528, −9.674843805778890730739940667370, −8.172069290789149577081904912736, −7.74336571411823674553675623446, −6.52702342475106697057035998705, −5.42259091252602543399676506535, −4.28635604587339142512595463752, −3.31909427716400977595329950904, −1.53250749395374940248601137918, −0.30530277897190286287322627838,
1.89323932960824113722280570522, 4.11183405646833851038951537448, 5.04478706109218475816428740978, 5.77411546228479850358406786218, 6.69333268895802837335000645617, 7.40176782462339955121016732591, 8.776628857306289954858924093555, 9.371756153318049470951402751851, 10.05532865824470956739212636739, 11.20172626789545083682998168379