L(s) = 1 | − 2-s + 4-s − 3.55·5-s − 0.319·7-s − 8-s + 3.55·10-s − 4.45·11-s − 4.67·13-s + 0.319·14-s + 16-s − 1.13·17-s + 1.85·19-s − 3.55·20-s + 4.45·22-s − 0.144·23-s + 7.62·25-s + 4.67·26-s − 0.319·28-s + 4.35·29-s + 0.671·31-s − 32-s + 1.13·34-s + 1.13·35-s + 7.58·37-s − 1.85·38-s + 3.55·40-s − 2.17·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.58·5-s − 0.120·7-s − 0.353·8-s + 1.12·10-s − 1.34·11-s − 1.29·13-s + 0.0853·14-s + 0.250·16-s − 0.275·17-s + 0.426·19-s − 0.794·20-s + 0.949·22-s − 0.0300·23-s + 1.52·25-s + 0.917·26-s − 0.0603·28-s + 0.809·29-s + 0.120·31-s − 0.176·32-s + 0.194·34-s + 0.191·35-s + 1.24·37-s − 0.301·38-s + 0.561·40-s − 0.340·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4650223005\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4650223005\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 0.319T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 + 0.144T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 0.671T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 0.805T + 53T^{2} \) |
| 59 | \( 1 + 2.98T + 59T^{2} \) |
| 61 | \( 1 + 3.49T + 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 5.41T + 83T^{2} \) |
| 89 | \( 1 + 5.05T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−9.558196541543574692816247480180, −8.497728255078483243975689389927, −7.85732932540023431619056362822, −7.45107466017986045335870640250, −6.58327255139859330050299519655, −5.22967486476833084875099420213, −4.48143383017791727258662275160, −3.27353914878062847376189962831, −2.43104033222191600895578815673, −0.51071299167289054542458070736,
0.51071299167289054542458070736, 2.43104033222191600895578815673, 3.27353914878062847376189962831, 4.48143383017791727258662275160, 5.22967486476833084875099420213, 6.58327255139859330050299519655, 7.45107466017986045335870640250, 7.85732932540023431619056362822, 8.497728255078483243975689389927, 9.558196541543574692816247480180