| L(s) = 1 | − 2-s + 1.61·3-s + 4-s + 5-s − 1.61·6-s − 7-s − 8-s − 0.381·9-s − 10-s + 1.61·12-s − 5.61·13-s + 14-s + 1.61·15-s + 16-s − 4.61·17-s + 0.381·18-s + 5.23·19-s + 20-s − 1.61·21-s + 8.47·23-s − 1.61·24-s + 25-s + 5.61·26-s − 5.47·27-s − 28-s + 5.85·29-s − 1.61·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.934·3-s + 0.5·4-s + 0.447·5-s − 0.660·6-s − 0.377·7-s − 0.353·8-s − 0.127·9-s − 0.316·10-s + 0.467·12-s − 1.55·13-s + 0.267·14-s + 0.417·15-s + 0.250·16-s − 1.12·17-s + 0.0900·18-s + 1.20·19-s + 0.223·20-s − 0.353·21-s + 1.76·23-s − 0.330·24-s + 0.200·25-s + 1.10·26-s − 1.05·27-s − 0.188·28-s + 1.08·29-s − 0.295·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + 7.32T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 - 0.854T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 0.291T + 89T^{2} \) |
| 97 | \( 1 + 3.38T + 97T^{2} \) |
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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
−7.39818965370076322839525424556, −6.98899408251584528600183720536, −6.36857887485927252639699776255, −5.21723646533143036010134286238, −4.83139705307050625500049062569, −3.50139610773278868663914581379, −2.79530560741138540366489672004, −2.41156282375535625445918186100, −1.30736366624602087019287236653, 0,
1.30736366624602087019287236653, 2.41156282375535625445918186100, 2.79530560741138540366489672004, 3.50139610773278868663914581379, 4.83139705307050625500049062569, 5.21723646533143036010134286238, 6.36857887485927252639699776255, 6.98899408251584528600183720536, 7.39818965370076322839525424556
