L(s) = 1 | + 2-s + 4-s + 2.56·5-s + 2.27·7-s + 8-s + 2.56·10-s − 0.680·11-s + 3.64·13-s + 2.27·14-s + 16-s − 1.93·17-s − 1.65·19-s + 2.56·20-s − 0.680·22-s + 6.35·23-s + 1.57·25-s + 3.64·26-s + 2.27·28-s + 8.02·29-s − 0.995·31-s + 32-s − 1.93·34-s + 5.84·35-s − 8.32·37-s − 1.65·38-s + 2.56·40-s − 2.25·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.14·5-s + 0.861·7-s + 0.353·8-s + 0.811·10-s − 0.205·11-s + 1.01·13-s + 0.609·14-s + 0.250·16-s − 0.470·17-s − 0.380·19-s + 0.573·20-s − 0.145·22-s + 1.32·23-s + 0.315·25-s + 0.714·26-s + 0.430·28-s + 1.49·29-s − 0.178·31-s + 0.176·32-s − 0.332·34-s + 0.988·35-s − 1.36·37-s − 0.269·38-s + 0.405·40-s − 0.352·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.527869866\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.527869866\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 11 | \( 1 + 0.680T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 - 8.02T + 29T^{2} \) |
| 31 | \( 1 + 0.995T + 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 43 | \( 1 - 0.273T + 43T^{2} \) |
| 47 | \( 1 - 2.71T + 47T^{2} \) |
| 53 | \( 1 + 7.97T + 53T^{2} \) |
| 59 | \( 1 + 3.85T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 + 6.56T + 71T^{2} \) |
| 73 | \( 1 + 2.15T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 - 18.7T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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
−8.525836153732025918667275882898, −7.66610408181324913921358862827, −6.61560314580236508662492793573, −6.30614535733977142313122968250, −5.23978537688594681432231376619, −4.95672848930540506992317057567, −3.91115971124331279030040975628, −2.91921638087125799684765172946, −2.01773756270550947080025026308, −1.23403738209583333744408881866,
1.23403738209583333744408881866, 2.01773756270550947080025026308, 2.91921638087125799684765172946, 3.91115971124331279030040975628, 4.95672848930540506992317057567, 5.23978537688594681432231376619, 6.30614535733977142313122968250, 6.61560314580236508662492793573, 7.66610408181324913921358862827, 8.525836153732025918667275882898