| L(s) = 1 | + 2·7-s − 2.64·11-s − 5.29·17-s − 19-s + 5.29·23-s − 5·25-s + 7.93·29-s − 3·31-s − 8·37-s + 2.64·41-s + 6·43-s + 13.2·47-s − 3·49-s + 13.2·53-s − 10.5·59-s − 13·61-s + 3·67-s − 5.29·71-s − 11·73-s − 5.29·77-s − 79-s + 2.64·83-s − 13.2·89-s − 16·97-s − 10.5·101-s + 10.5·107-s + 2·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.797·11-s − 1.28·17-s − 0.229·19-s + 1.10·23-s − 25-s + 1.47·29-s − 0.538·31-s − 1.31·37-s + 0.413·41-s + 0.914·43-s + 1.92·47-s − 0.428·49-s + 1.81·53-s − 1.37·59-s − 1.66·61-s + 0.366·67-s − 0.627·71-s − 1.28·73-s − 0.603·77-s − 0.112·79-s + 0.290·83-s − 1.40·89-s − 1.62·97-s − 1.05·101-s + 1.02·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 + 5.29T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−7.41893275930779634012895651473, −6.94107150428303233606677347801, −6.00056980025883650870536488212, −5.36774722842709610585634076926, −4.60323850176097394322477579560, −4.10062717373090040920590038846, −2.91601255266804851984608950242, −2.28727431571152410399892603527, −1.30529084212675840866238653414, 0,
1.30529084212675840866238653414, 2.28727431571152410399892603527, 2.91601255266804851984608950242, 4.10062717373090040920590038846, 4.60323850176097394322477579560, 5.36774722842709610585634076926, 6.00056980025883650870536488212, 6.94107150428303233606677347801, 7.41893275930779634012895651473
