| L(s) = 1 | − 4.37·5-s − 7-s + 0.372·11-s − 13-s − 0.372·17-s + 1.62·19-s + 4.37·23-s + 14.1·25-s − 10.3·29-s + 0.744·31-s + 4.37·35-s − 0.372·37-s + 6·41-s + 1.62·43-s + 13.4·47-s + 49-s − 8.74·53-s − 1.62·55-s − 8.74·59-s + 5.11·61-s + 4.37·65-s + 12·67-s − 10·71-s + 3.62·73-s − 0.372·77-s − 3.25·79-s + 12.7·83-s + ⋯ |
| L(s) = 1 | − 1.95·5-s − 0.377·7-s + 0.112·11-s − 0.277·13-s − 0.0902·17-s + 0.373·19-s + 0.911·23-s + 2.82·25-s − 1.92·29-s + 0.133·31-s + 0.739·35-s − 0.0612·37-s + 0.937·41-s + 0.248·43-s + 1.96·47-s + 0.142·49-s − 1.20·53-s − 0.219·55-s − 1.13·59-s + 0.655·61-s + 0.542·65-s + 1.46·67-s − 1.18·71-s + 0.424·73-s − 0.0424·77-s − 0.366·79-s + 1.39·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 - 0.372T + 11T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.744T + 31T^{2} \) |
| 37 | \( 1 + 0.372T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 8.74T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 + 3.25T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 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
−7.51697618624304333324865427926, −7.26505107913210410167804153663, −6.39936279073400702802731750378, −5.40459195550528532357407695594, −4.63772701242220749168725415090, −3.89386391665239484029229981933, −3.40352900719775832249539485296, −2.50859500788490083314487380344, −1.00052919363521109336459018594, 0,
1.00052919363521109336459018594, 2.50859500788490083314487380344, 3.40352900719775832249539485296, 3.89386391665239484029229981933, 4.63772701242220749168725415090, 5.40459195550528532357407695594, 6.39936279073400702802731750378, 7.26505107913210410167804153663, 7.51697618624304333324865427926
