L(s) = 1 | − 2-s + 3-s + 4-s + 2.48·5-s − 6-s + 1.57·7-s − 8-s + 9-s − 2.48·10-s − 11-s + 12-s − 4.97·13-s − 1.57·14-s + 2.48·15-s + 16-s + 7.22·17-s − 18-s + 5.46·19-s + 2.48·20-s + 1.57·21-s + 22-s − 1.75·23-s − 24-s + 1.16·25-s + 4.97·26-s + 27-s + 1.57·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.11·5-s − 0.408·6-s + 0.596·7-s − 0.353·8-s + 0.333·9-s − 0.784·10-s − 0.301·11-s + 0.288·12-s − 1.37·13-s − 0.421·14-s + 0.640·15-s + 0.250·16-s + 1.75·17-s − 0.235·18-s + 1.25·19-s + 0.555·20-s + 0.344·21-s + 0.213·22-s − 0.366·23-s − 0.204·24-s + 0.232·25-s + 0.975·26-s + 0.192·27-s + 0.298·28-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
\(L(1)\) |
\(\approx\) |
\(2.375859662\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.375859662\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.48T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 0.0841T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 + 5.61T + 47T^{2} \) |
| 53 | \( 1 - 0.738T + 53T^{2} \) |
| 59 | \( 1 - 6.61T + 59T^{2} \) |
| 67 | \( 1 - 7.08T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 0.359T + 83T^{2} \) |
| 89 | \( 1 - 4.63T + 89T^{2} \) |
| 97 | \( 1 - 5.94T + 97T^{2} \) |
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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
−8.349902830216676724111823939326, −7.74891798410272132053523653488, −7.33214473089070579099038545505, −6.30286045391391046598228483177, −5.40074361545595361970594996761, −4.97673425380646067391230394658, −3.56019868620478195573989182361, −2.66471693224979398723133324248, −1.95961061094092407107240920579, −1.00241819620386381406635628560,
1.00241819620386381406635628560, 1.95961061094092407107240920579, 2.66471693224979398723133324248, 3.56019868620478195573989182361, 4.97673425380646067391230394658, 5.40074361545595361970594996761, 6.30286045391391046598228483177, 7.33214473089070579099038545505, 7.74891798410272132053523653488, 8.349902830216676724111823939326