L(s) = 1 | − 2-s + 3.30·3-s + 4-s − 5-s − 3.30·6-s + 0.302·7-s − 8-s + 7.90·9-s + 10-s + 5.30·11-s + 3.30·12-s − 0.302·13-s − 0.302·14-s − 3.30·15-s + 16-s + 3.90·17-s − 7.90·18-s + 4.90·19-s − 20-s + 1.00·21-s − 5.30·22-s − 3.30·24-s + 25-s + 0.302·26-s + 16.2·27-s + 0.302·28-s + 4.60·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.90·3-s + 0.5·4-s − 0.447·5-s − 1.34·6-s + 0.114·7-s − 0.353·8-s + 2.63·9-s + 0.316·10-s + 1.59·11-s + 0.953·12-s − 0.0839·13-s − 0.0809·14-s − 0.852·15-s + 0.250·16-s + 0.947·17-s − 1.86·18-s + 1.12·19-s − 0.223·20-s + 0.218·21-s − 1.13·22-s − 0.674·24-s + 0.200·25-s + 0.0593·26-s + 3.11·27-s + 0.0572·28-s + 0.855·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.505854253\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.505854253\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 - 0.302T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 + 0.302T + 13T^{2} \) |
| 17 | \( 1 - 3.90T + 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 2.90T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 6.51T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2.69T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.248573402024787708120180275294, −7.76724255537707758042311995469, −6.98661838218681157121995456683, −6.56884801584253344478848150125, −5.13026902404799126415466140273, −4.14976797084711345520746040983, −3.39399312264718667133158518062, −2.99727562464875189347740904144, −1.73148530922489940735699376701, −1.17774502232057396099768266948,
1.17774502232057396099768266948, 1.73148530922489940735699376701, 2.99727562464875189347740904144, 3.39399312264718667133158518062, 4.14976797084711345520746040983, 5.13026902404799126415466140273, 6.56884801584253344478848150125, 6.98661838218681157121995456683, 7.76724255537707758042311995469, 8.248573402024787708120180275294