| L(s) = 1 | − 2-s − 3-s + 4-s + 4.37·5-s + 6-s + 0.258·7-s − 8-s + 9-s − 4.37·10-s + 1.21·11-s − 12-s − 4.93·13-s − 0.258·14-s − 4.37·15-s + 16-s − 0.541·17-s − 18-s − 2.22·19-s + 4.37·20-s − 0.258·21-s − 1.21·22-s − 6.97·23-s + 24-s + 14.1·25-s + 4.93·26-s − 27-s + 0.258·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.95·5-s + 0.408·6-s + 0.0977·7-s − 0.353·8-s + 0.333·9-s − 1.38·10-s + 0.364·11-s − 0.288·12-s − 1.36·13-s − 0.0691·14-s − 1.12·15-s + 0.250·16-s − 0.131·17-s − 0.235·18-s − 0.511·19-s + 0.978·20-s − 0.0564·21-s − 0.258·22-s − 1.45·23-s + 0.204·24-s + 2.82·25-s + 0.968·26-s − 0.192·27-s + 0.0488·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 - 4.37T + 5T^{2} \) |
| 7 | \( 1 - 0.258T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 + 0.541T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 8.87T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 + 7.55T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 + 7.93T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 - 3.96T + 79T^{2} \) |
| 83 | \( 1 - 6.64T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 6.77T + 97T^{2} \) |
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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
−7.64224045904847884862717583969, −6.99152946850990005092739657439, −6.28668999198671319525898999658, −5.76883001271487600622359166688, −5.14051141129518301422276343510, −4.24055793176720096002031442134, −2.82181133072345515089299881646, −2.04240992955785183721527701930, −1.48431846871649353012192641622, 0,
1.48431846871649353012192641622, 2.04240992955785183721527701930, 2.82181133072345515089299881646, 4.24055793176720096002031442134, 5.14051141129518301422276343510, 5.76883001271487600622359166688, 6.28668999198671319525898999658, 6.99152946850990005092739657439, 7.64224045904847884862717583969
