L(s) = 1 | + 2.70·2-s − 0.991·3-s + 5.33·4-s − 5-s − 2.68·6-s + 9.03·8-s − 2.01·9-s − 2.70·10-s + 0.218·11-s − 5.28·12-s − 4.12·13-s + 0.991·15-s + 13.7·16-s − 3.34·17-s − 5.46·18-s + 0.0746·19-s − 5.33·20-s + 0.590·22-s − 4.04·23-s − 8.95·24-s + 25-s − 11.1·26-s + 4.97·27-s + 4.71·29-s + 2.68·30-s + 3.30·31-s + 19.2·32-s + ⋯ |
L(s) = 1 | + 1.91·2-s − 0.572·3-s + 2.66·4-s − 0.447·5-s − 1.09·6-s + 3.19·8-s − 0.672·9-s − 0.856·10-s + 0.0657·11-s − 1.52·12-s − 1.14·13-s + 0.256·15-s + 3.44·16-s − 0.812·17-s − 1.28·18-s + 0.0171·19-s − 1.19·20-s + 0.125·22-s − 0.843·23-s − 1.82·24-s + 0.200·25-s − 2.18·26-s + 0.957·27-s + 0.875·29-s + 0.490·30-s + 0.593·31-s + 3.40·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 3 | \( 1 + 0.991T + 3T^{2} \) |
| 11 | \( 1 - 0.218T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 - 0.0746T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 - 4.71T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 - 2.31T + 43T^{2} \) |
| 47 | \( 1 + 4.52T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 3.15T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 8.86T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 + 4.67T + 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−7.03305313389827156799000219509, −6.42137087374215911720358044535, −5.95955699054364735461501350135, −5.11633386061919446769223383858, −4.67589009148778068875502944031, −4.11616062988015548607637684005, −3.07459661563636943950468257847, −2.66190222352062315890695983029, −1.63202452804487345790937152617, 0,
1.63202452804487345790937152617, 2.66190222352062315890695983029, 3.07459661563636943950468257847, 4.11616062988015548607637684005, 4.67589009148778068875502944031, 5.11633386061919446769223383858, 5.95955699054364735461501350135, 6.42137087374215911720358044535, 7.03305313389827156799000219509