| L(s) = 1 | − 3.37·5-s + 7-s − 2·11-s − 13-s + 0.627·19-s + 1.37·23-s + 6.37·25-s − 1.37·29-s − 3.37·31-s − 3.37·35-s − 4.74·37-s + 2.74·41-s − 6.11·43-s + 0.627·47-s + 49-s − 5.37·53-s + 6.74·55-s + 8·59-s + 6·61-s + 3.37·65-s + 1.25·67-s + 8.74·71-s + 14.8·73-s − 2·77-s − 6.11·79-s + 6.11·83-s + 7.37·89-s + ⋯ |
| L(s) = 1 | − 1.50·5-s + 0.377·7-s − 0.603·11-s − 0.277·13-s + 0.144·19-s + 0.286·23-s + 1.27·25-s − 0.254·29-s − 0.605·31-s − 0.570·35-s − 0.780·37-s + 0.428·41-s − 0.932·43-s + 0.0915·47-s + 0.142·49-s − 0.737·53-s + 0.909·55-s + 1.04·59-s + 0.768·61-s + 0.418·65-s + 0.153·67-s + 1.03·71-s + 1.73·73-s − 0.227·77-s − 0.688·79-s + 0.671·83-s + 0.781·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030913093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030913093\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.37T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 6.11T + 43T^{2} \) |
| 47 | \( 1 - 0.627T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 1.25T + 67T^{2} \) |
| 71 | \( 1 - 8.74T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 - 6.11T + 83T^{2} \) |
| 89 | \( 1 - 7.37T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.441080468982545778789576954241, −7.88724769658825201684170942906, −7.33483138108927611865025647964, −6.59827426122229118522192198205, −5.37543672226025322996748886309, −4.81186127008521047232311558845, −3.88683454700434800306862288174, −3.24536642018500696225430234400, −2.07983895515143152664770228243, −0.59191358546458505685976335368,
0.59191358546458505685976335368, 2.07983895515143152664770228243, 3.24536642018500696225430234400, 3.88683454700434800306862288174, 4.81186127008521047232311558845, 5.37543672226025322996748886309, 6.59827426122229118522192198205, 7.33483138108927611865025647964, 7.88724769658825201684170942906, 8.441080468982545778789576954241
