| L(s) = 1 | + 2.58·3-s + 1.24·5-s + 0.899·7-s + 3.68·9-s − 3.77·11-s − 4.29·13-s + 3.22·15-s − 7.86·17-s + 6.61·19-s + 2.32·21-s − 3.37·23-s − 3.44·25-s + 1.78·27-s − 2.81·29-s + 1.72·31-s − 9.77·33-s + 1.12·35-s − 4.86·37-s − 11.0·39-s + 5.86·41-s − 7.30·43-s + 4.60·45-s − 2.08·47-s − 6.19·49-s − 20.3·51-s + 3.26·53-s − 4.71·55-s + ⋯ |
| L(s) = 1 | + 1.49·3-s + 0.557·5-s + 0.339·7-s + 1.22·9-s − 1.13·11-s − 1.18·13-s + 0.832·15-s − 1.90·17-s + 1.51·19-s + 0.507·21-s − 0.704·23-s − 0.688·25-s + 0.343·27-s − 0.522·29-s + 0.309·31-s − 1.70·33-s + 0.189·35-s − 0.800·37-s − 1.77·39-s + 0.915·41-s − 1.11·43-s + 0.685·45-s − 0.303·47-s − 0.884·49-s − 2.84·51-s + 0.448·53-s − 0.635·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 - 2.58T + 3T^{2} \) |
| 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 - 0.899T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 + 7.86T + 17T^{2} \) |
| 19 | \( 1 - 6.61T + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 + 2.81T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 + 4.86T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 + 7.30T + 43T^{2} \) |
| 47 | \( 1 + 2.08T + 47T^{2} \) |
| 53 | \( 1 - 3.26T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 5.15T + 79T^{2} \) |
| 83 | \( 1 + 6.55T + 83T^{2} \) |
| 89 | \( 1 + 4.60T + 89T^{2} \) |
| 97 | \( 1 - 9.16T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.35247681499885637775424386220, −7.19292232476993109859296385330, −6.00009455252148877191122634585, −5.21692644442472617856528714793, −4.60069615445324881668897884817, −3.72627394984848124758363224289, −2.81534757150352532494601507899, −2.31620302242785855634926608776, −1.73076610137598093076888306731, 0,
1.73076610137598093076888306731, 2.31620302242785855634926608776, 2.81534757150352532494601507899, 3.72627394984848124758363224289, 4.60069615445324881668897884817, 5.21692644442472617856528714793, 6.00009455252148877191122634585, 7.19292232476993109859296385330, 7.35247681499885637775424386220
