| L(s) = 1 | − 2-s − 3-s + 4-s − 1.40·5-s + 6-s + 2.64·7-s − 8-s + 9-s + 1.40·10-s + 11-s − 12-s − 2.22·13-s − 2.64·14-s + 1.40·15-s + 16-s − 2.42·17-s − 18-s + 1.90·19-s − 1.40·20-s − 2.64·21-s − 22-s − 0.204·23-s + 24-s − 3.03·25-s + 2.22·26-s − 27-s + 2.64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.627·5-s + 0.408·6-s + 0.998·7-s − 0.353·8-s + 0.333·9-s + 0.443·10-s + 0.301·11-s − 0.288·12-s − 0.615·13-s − 0.705·14-s + 0.362·15-s + 0.250·16-s − 0.588·17-s − 0.235·18-s + 0.437·19-s − 0.313·20-s − 0.576·21-s − 0.213·22-s − 0.0427·23-s + 0.204·24-s − 0.606·25-s + 0.435·26-s − 0.192·27-s + 0.499·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 + 0.204T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 - 9.76T + 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 + 7.76T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 9.17T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 67 | \( 1 - 6.95T + 67T^{2} \) |
| 71 | \( 1 + 7.05T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 3.02T + 97T^{2} \) |
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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
−8.119719266815137991801179724366, −7.42175320442898137865486363281, −6.79621668805286971914667286972, −5.96883973776609693178071164736, −4.97892389119316920087372770282, −4.46168188376189661945895024576, −3.38721095397078897541326268346, −2.19552113860245821206072147239, −1.23882970412425720108910095206, 0,
1.23882970412425720108910095206, 2.19552113860245821206072147239, 3.38721095397078897541326268346, 4.46168188376189661945895024576, 4.97892389119316920087372770282, 5.96883973776609693178071164736, 6.79621668805286971914667286972, 7.42175320442898137865486363281, 8.119719266815137991801179724366
