L(s) = 1 | + 2-s − 0.765·3-s + 4-s − 5-s − 0.765·6-s + 2.95·7-s + 8-s − 2.41·9-s − 10-s − 11-s − 0.765·12-s − 2.75·13-s + 2.95·14-s + 0.765·15-s + 16-s + 0.534·17-s − 2.41·18-s + 6.63·19-s − 20-s − 2.26·21-s − 22-s − 6.23·23-s − 0.765·24-s + 25-s − 2.75·26-s + 4.14·27-s + 2.95·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.441·3-s + 0.5·4-s − 0.447·5-s − 0.312·6-s + 1.11·7-s + 0.353·8-s − 0.804·9-s − 0.316·10-s − 0.301·11-s − 0.220·12-s − 0.763·13-s + 0.790·14-s + 0.197·15-s + 0.250·16-s + 0.129·17-s − 0.569·18-s + 1.52·19-s − 0.223·20-s − 0.493·21-s − 0.213·22-s − 1.30·23-s − 0.156·24-s + 0.200·25-s − 0.540·26-s + 0.797·27-s + 0.559·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.430278607\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.430278607\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 + 0.765T + 3T^{2} \) |
| 7 | \( 1 - 2.95T + 7T^{2} \) |
| 13 | \( 1 + 2.75T + 13T^{2} \) |
| 17 | \( 1 - 0.534T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 0.857T + 31T^{2} \) |
| 37 | \( 1 - 7.28T + 37T^{2} \) |
| 41 | \( 1 - 7.08T + 41T^{2} \) |
| 43 | \( 1 - 8.08T + 43T^{2} \) |
| 47 | \( 1 + 2.78T + 47T^{2} \) |
| 53 | \( 1 + 5.77T + 53T^{2} \) |
| 59 | \( 1 + 5.27T + 59T^{2} \) |
| 61 | \( 1 - 2.10T + 61T^{2} \) |
| 67 | \( 1 + 3.63T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 79 | \( 1 - 9.85T + 79T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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
−7.77144505337483483391588687386, −7.28981020666279027418083686641, −6.16178354823191983587412843421, −5.65684590479583039018442499613, −5.02319916283556903661134492819, −4.47840234180796269690843245894, −3.59400445811978452828712357197, −2.74200888261616494859248417868, −1.92636314371433718747645005413, −0.69888702259428413170448410024,
0.69888702259428413170448410024, 1.92636314371433718747645005413, 2.74200888261616494859248417868, 3.59400445811978452828712357197, 4.47840234180796269690843245894, 5.02319916283556903661134492819, 5.65684590479583039018442499613, 6.16178354823191983587412843421, 7.28981020666279027418083686641, 7.77144505337483483391588687386