L(s) = 1 | − 1.67·2-s − 1.48·3-s + 0.810·4-s − 5-s + 2.49·6-s + 7-s + 1.99·8-s − 0.792·9-s + 1.67·10-s − 3.82·11-s − 1.20·12-s − 6.09·13-s − 1.67·14-s + 1.48·15-s − 4.96·16-s − 2.99·17-s + 1.32·18-s − 3.51·19-s − 0.810·20-s − 1.48·21-s + 6.41·22-s − 3.88·23-s − 2.96·24-s + 25-s + 10.2·26-s + 5.63·27-s + 0.810·28-s + ⋯ |
L(s) = 1 | − 1.18·2-s − 0.857·3-s + 0.405·4-s − 0.447·5-s + 1.01·6-s + 0.377·7-s + 0.704·8-s − 0.264·9-s + 0.530·10-s − 1.15·11-s − 0.347·12-s − 1.68·13-s − 0.448·14-s + 0.383·15-s − 1.24·16-s − 0.727·17-s + 0.313·18-s − 0.807·19-s − 0.181·20-s − 0.324·21-s + 1.36·22-s − 0.809·23-s − 0.604·24-s + 0.200·25-s + 2.00·26-s + 1.08·27-s + 0.153·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 3 | \( 1 + 1.48T + 3T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 + 2.99T + 17T^{2} \) |
| 19 | \( 1 + 3.51T + 19T^{2} \) |
| 23 | \( 1 + 3.88T + 23T^{2} \) |
| 29 | \( 1 - 6.72T + 29T^{2} \) |
| 31 | \( 1 - 2.22T + 31T^{2} \) |
| 37 | \( 1 + 8.02T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 - 9.86T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 - 3.02T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 7.95T + 61T^{2} \) |
| 67 | \( 1 - 5.74T + 67T^{2} \) |
| 71 | \( 1 + 2.42T + 71T^{2} \) |
| 73 | \( 1 - 9.64T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 - 9.76T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 + 1.99T + 97T^{2} \) |
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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
−7.58943040263391276707783770302, −7.05539510273216607231147150918, −6.24356910021176522858530573372, −5.30503022774064884523135043941, −4.76603818906696832815097252514, −4.20836006774541995140162482554, −2.66773871806240046733875652310, −2.15641019031374933551244836833, −0.68907504271414041266448592329, 0,
0.68907504271414041266448592329, 2.15641019031374933551244836833, 2.66773871806240046733875652310, 4.20836006774541995140162482554, 4.76603818906696832815097252514, 5.30503022774064884523135043941, 6.24356910021176522858530573372, 7.05539510273216607231147150918, 7.58943040263391276707783770302