L(s) = 1 | − 2.59·2-s + 3-s + 4.72·4-s + 3.19·5-s − 2.59·6-s − 1.98·7-s − 7.07·8-s + 9-s − 8.27·10-s + 5.06·11-s + 4.72·12-s + 0.245·13-s + 5.15·14-s + 3.19·15-s + 8.89·16-s + 4.87·17-s − 2.59·18-s + 0.823·19-s + 15.0·20-s − 1.98·21-s − 13.1·22-s + 1.93·23-s − 7.07·24-s + 5.17·25-s − 0.637·26-s + 27-s − 9.39·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 0.577·3-s + 2.36·4-s + 1.42·5-s − 1.05·6-s − 0.750·7-s − 2.50·8-s + 0.333·9-s − 2.61·10-s + 1.52·11-s + 1.36·12-s + 0.0681·13-s + 1.37·14-s + 0.823·15-s + 2.22·16-s + 1.18·17-s − 0.611·18-s + 0.188·19-s + 3.37·20-s − 0.433·21-s − 2.80·22-s + 0.402·23-s − 1.44·24-s + 1.03·25-s − 0.125·26-s + 0.192·27-s − 1.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.703182626\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.703182626\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 0.245T + 13T^{2} \) |
| 17 | \( 1 - 4.87T + 17T^{2} \) |
| 19 | \( 1 - 0.823T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 3.64T + 31T^{2} \) |
| 37 | \( 1 + 4.19T + 37T^{2} \) |
| 41 | \( 1 - 9.38T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 0.500T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 + 7.13T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 8.14T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 + 3.53T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−7.960050318425341006299697311605, −7.23922848601649960313232695974, −6.69678104235909520702028652222, −6.06733021788164282550587339431, −5.49692711138828385341472356340, −3.95274492886725161659817412130, −3.11021288774008379426685071309, −2.31942438750072458627733500338, −1.54229714638056911949710060672, −0.902796633032825307175650793160,
0.902796633032825307175650793160, 1.54229714638056911949710060672, 2.31942438750072458627733500338, 3.11021288774008379426685071309, 3.95274492886725161659817412130, 5.49692711138828385341472356340, 6.06733021788164282550587339431, 6.69678104235909520702028652222, 7.23922848601649960313232695974, 7.960050318425341006299697311605