L(s) = 1 | + 0.528·2-s − 0.931·3-s − 1.72·4-s − 0.425·5-s − 0.492·6-s − 2.15·7-s − 1.96·8-s − 2.13·9-s − 0.225·10-s − 2.72·11-s + 1.60·12-s + 4.70·13-s − 1.14·14-s + 0.396·15-s + 2.40·16-s − 1.13·17-s − 1.12·18-s − 4.22·19-s + 0.732·20-s + 2.01·21-s − 1.43·22-s − 0.880·23-s + 1.83·24-s − 4.81·25-s + 2.48·26-s + 4.78·27-s + 3.71·28-s + ⋯ |
L(s) = 1 | + 0.373·2-s − 0.537·3-s − 0.860·4-s − 0.190·5-s − 0.201·6-s − 0.815·7-s − 0.695·8-s − 0.710·9-s − 0.0711·10-s − 0.820·11-s + 0.462·12-s + 1.30·13-s − 0.304·14-s + 0.102·15-s + 0.600·16-s − 0.274·17-s − 0.265·18-s − 0.970·19-s + 0.163·20-s + 0.438·21-s − 0.306·22-s − 0.183·23-s + 0.374·24-s − 0.963·25-s + 0.488·26-s + 0.920·27-s + 0.701·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6134242613\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6134242613\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 - 0.528T + 2T^{2} \) |
| 3 | \( 1 + 0.931T + 3T^{2} \) |
| 5 | \( 1 + 0.425T + 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 + 4.22T + 19T^{2} \) |
| 23 | \( 1 + 0.880T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 0.386T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 - 2.51T + 41T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
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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
−9.201769529966626096329480107791, −8.516000524670741579583338937106, −7.81375816412588395578724603751, −6.56455225118245557599766202797, −5.80696981931806367208218637236, −5.44111800341920429918601560961, −4.14839674278497143866894053886, −3.63310834268346304406417503972, −2.45225956876735470913154513434, −0.49317088632185569576052197214,
0.49317088632185569576052197214, 2.45225956876735470913154513434, 3.63310834268346304406417503972, 4.14839674278497143866894053886, 5.44111800341920429918601560961, 5.80696981931806367208218637236, 6.56455225118245557599766202797, 7.81375816412588395578724603751, 8.516000524670741579583338937106, 9.201769529966626096329480107791