L(s) = 1 | − 2-s − 2.26·3-s + 4-s + 5-s + 2.26·6-s + 1.24·7-s − 8-s + 2.11·9-s − 10-s − 11-s − 2.26·12-s + 0.215·13-s − 1.24·14-s − 2.26·15-s + 16-s − 4.59·17-s − 2.11·18-s − 2.16·19-s + 20-s − 2.82·21-s + 22-s + 1.42·23-s + 2.26·24-s + 25-s − 0.215·26-s + 2.00·27-s + 1.24·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.30·3-s + 0.5·4-s + 0.447·5-s + 0.923·6-s + 0.471·7-s − 0.353·8-s + 0.703·9-s − 0.316·10-s − 0.301·11-s − 0.652·12-s + 0.0597·13-s − 0.333·14-s − 0.583·15-s + 0.250·16-s − 1.11·17-s − 0.497·18-s − 0.496·19-s + 0.223·20-s − 0.615·21-s + 0.213·22-s + 0.297·23-s + 0.461·24-s + 0.200·25-s − 0.0422·26-s + 0.386·27-s + 0.235·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6476321582\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6476321582\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 13 | \( 1 - 0.215T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 - 5.14T + 41T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 53 | \( 1 + 0.324T + 53T^{2} \) |
| 59 | \( 1 - 0.248T + 59T^{2} \) |
| 61 | \( 1 - 0.487T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 + 0.463T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 0.295T + 97T^{2} \) |
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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
−8.325890966674251803508528262972, −7.49771707274327564085368206398, −6.77919653129666267114463614516, −6.15943573430467095883238753008, −5.48242824611052497180732560535, −4.85824457399105729195367078813, −3.89575651395457560349448038395, −2.52692939246473177063689615272, −1.70725619319663960912269870228, −0.52662309714559379852450488702,
0.52662309714559379852450488702, 1.70725619319663960912269870228, 2.52692939246473177063689615272, 3.89575651395457560349448038395, 4.85824457399105729195367078813, 5.48242824611052497180732560535, 6.15943573430467095883238753008, 6.77919653129666267114463614516, 7.49771707274327564085368206398, 8.325890966674251803508528262972