| L(s) = 1 | + 3-s − 0.551·5-s − 7-s + 9-s + 0.103·11-s + 5.69·13-s − 0.551·15-s − 4.24·17-s − 19-s − 21-s − 23-s − 4.69·25-s + 27-s + 1.14·29-s − 6.24·31-s + 0.103·33-s + 0.551·35-s + 3.35·37-s + 5.69·39-s − 9.28·41-s + 0.799·43-s − 0.551·45-s − 8.14·47-s + 49-s − 4.24·51-s + 0.695·53-s − 0.0573·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.246·5-s − 0.377·7-s + 0.333·9-s + 0.0313·11-s + 1.57·13-s − 0.142·15-s − 1.03·17-s − 0.229·19-s − 0.218·21-s − 0.208·23-s − 0.939·25-s + 0.192·27-s + 0.212·29-s − 1.12·31-s + 0.0180·33-s + 0.0932·35-s + 0.550·37-s + 0.911·39-s − 1.45·41-s + 0.121·43-s − 0.0822·45-s − 1.18·47-s + 0.142·49-s − 0.594·51-s + 0.0955·53-s − 0.00772·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 0.551T + 5T^{2} \) |
| 11 | \( 1 - 0.103T + 11T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 + 9.28T + 41T^{2} \) |
| 43 | \( 1 - 0.799T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 - 0.695T + 53T^{2} \) |
| 59 | \( 1 - 0.695T + 59T^{2} \) |
| 61 | \( 1 - 7.55T + 61T^{2} \) |
| 67 | \( 1 - 0.551T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 + 8.45T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 - 9.69T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−7.60496406420858842997410705788, −6.76993923470952246257103141829, −6.27215051840628555631331855670, −5.49474836182993703351749928408, −4.46632165789793993488487958456, −3.81006397087721952126679609148, −3.27580504993202616754446560811, −2.22801481779099395791771375818, −1.40810761122548944621775598874, 0,
1.40810761122548944621775598874, 2.22801481779099395791771375818, 3.27580504993202616754446560811, 3.81006397087721952126679609148, 4.46632165789793993488487958456, 5.49474836182993703351749928408, 6.27215051840628555631331855670, 6.76993923470952246257103141829, 7.60496406420858842997410705788
