L(s) = 1 | − 2-s − 3-s + 4-s + 1.47·5-s + 6-s − 8-s + 9-s − 1.47·10-s + 5.70·11-s − 12-s + 6.35·13-s − 1.47·15-s + 16-s − 4.12·17-s − 18-s + 3.39·19-s + 1.47·20-s − 5.70·22-s + 23-s + 24-s − 2.81·25-s − 6.35·26-s − 27-s + 2.39·29-s + 1.47·30-s − 7.81·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.661·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.467·10-s + 1.71·11-s − 0.288·12-s + 1.76·13-s − 0.381·15-s + 0.250·16-s − 1.00·17-s − 0.235·18-s + 0.778·19-s + 0.330·20-s − 1.21·22-s + 0.208·23-s + 0.204·24-s − 0.562·25-s − 1.24·26-s − 0.192·27-s + 0.444·29-s + 0.270·30-s − 1.40·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.807070861\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.807070861\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 - 5.70T + 11T^{2} \) |
| 13 | \( 1 - 6.35T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 - 7.39T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 0.520T + 53T^{2} \) |
| 59 | \( 1 - 7.37T + 59T^{2} \) |
| 61 | \( 1 + 5.04T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 9.87T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 9.22T + 83T^{2} \) |
| 89 | \( 1 + 6.43T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.079155139669855780588075430644, −7.09918126000824778000800559818, −6.58297658305180275937937198713, −5.98061188978677228110869265017, −5.50960626237394540237307161896, −4.15295736986589486305135903156, −3.78019718096811064255586789381, −2.46691615361862362854079444892, −1.47013736584370937487735917441, −0.898983693318008273191383283157,
0.898983693318008273191383283157, 1.47013736584370937487735917441, 2.46691615361862362854079444892, 3.78019718096811064255586789381, 4.15295736986589486305135903156, 5.50960626237394540237307161896, 5.98061188978677228110869265017, 6.58297658305180275937937198713, 7.09918126000824778000800559818, 8.079155139669855780588075430644