| L(s) = 1 | + 2i·2-s − 2·4-s + 5·11-s − 4i·13-s − 4·16-s − 4i·17-s + 5·19-s + 10i·22-s − 6i·23-s + 8·26-s + 5·29-s − 9·31-s − 8i·32-s + 8·34-s − 10i·37-s + 10i·38-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s − 4-s + 1.50·11-s − 1.10i·13-s − 16-s − 0.970i·17-s + 1.14·19-s + 2.13i·22-s − 1.25i·23-s + 1.56·26-s + 0.928·29-s − 1.61·31-s − 1.41i·32-s + 1.37·34-s − 1.64i·37-s + 1.62i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.890112848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.890112848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 8iT - 53T^{2} \) |
| 59 | \( 1 - T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 - T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
| 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
−9.052150047314487167564381113354, −8.416621609205187749505237078390, −7.41230652305881932370041170001, −7.10461552715674291774889312507, −6.13681259531888009002934379624, −5.53869196918469484730253253699, −4.69640535634946212378317671111, −3.70273475350232943408782369464, −2.49273549885662812806547616181, −0.793629708182262923546600713212,
1.22901712178924335094102592129, 1.80871195297257303763879179492, 3.13986918992379447472209950127, 3.85317928540879815311252466217, 4.50504761785847529968900571770, 5.77776791828149324925887121240, 6.68591512574905174627733785393, 7.39113424670458149948219062461, 8.647564499423975792781250859877, 9.334098062684827383259916935914
