| L(s) = 1 | + 3-s + 5-s + 2.37·7-s + 9-s − 4·11-s − 6.74·13-s + 15-s − 0.372·17-s − 4·19-s + 2.37·21-s − 23-s + 25-s + 27-s + 0.372·29-s + 2.37·31-s − 4·33-s + 2.37·35-s + 0.372·37-s − 6.74·39-s + 9.11·41-s + 4·43-s + 45-s − 4.74·47-s − 1.37·49-s − 0.372·51-s − 4.37·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.896·7-s + 0.333·9-s − 1.20·11-s − 1.87·13-s + 0.258·15-s − 0.0902·17-s − 0.917·19-s + 0.517·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.0691·29-s + 0.426·31-s − 0.696·33-s + 0.400·35-s + 0.0612·37-s − 1.07·39-s + 1.42·41-s + 0.609·43-s + 0.149·45-s − 0.692·47-s − 0.196·49-s − 0.0521·51-s − 0.600·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 0.372T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 0.372T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4.74T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 9.62T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 6.74T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−7.72230339195156108613828989845, −7.41982244020680942381796800472, −6.38231271320155089277394467338, −5.51960741172169006785967583448, −4.73905266777381929583168295432, −4.38024840381912781999929670930, −2.90280104401534538810761550468, −2.44870495310897663107958276257, −1.62080009644254426371035028780, 0,
1.62080009644254426371035028780, 2.44870495310897663107958276257, 2.90280104401534538810761550468, 4.38024840381912781999929670930, 4.73905266777381929583168295432, 5.51960741172169006785967583448, 6.38231271320155089277394467338, 7.41982244020680942381796800472, 7.72230339195156108613828989845
