| L(s) = 1 | − 3-s − 1.29·5-s + 2.24·7-s + 9-s + 0.755·11-s − 0.267·13-s + 1.29·15-s + 3.97·17-s − 3.12·19-s − 2.24·21-s + 2.83·23-s − 3.31·25-s − 27-s − 9.08·29-s + 8.95·31-s − 0.755·33-s − 2.91·35-s − 9.45·37-s + 0.267·39-s + 4.41·41-s − 6.10·43-s − 1.29·45-s − 6.79·47-s − 1.95·49-s − 3.97·51-s + 4.27·53-s − 0.980·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.580·5-s + 0.848·7-s + 0.333·9-s + 0.227·11-s − 0.0741·13-s + 0.334·15-s + 0.964·17-s − 0.717·19-s − 0.489·21-s + 0.591·23-s − 0.663·25-s − 0.192·27-s − 1.68·29-s + 1.60·31-s − 0.131·33-s − 0.492·35-s − 1.55·37-s + 0.0428·39-s + 0.689·41-s − 0.930·43-s − 0.193·45-s − 0.991·47-s − 0.279·49-s − 0.556·51-s + 0.586·53-s − 0.132·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 1.29T + 5T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 - 0.755T + 11T^{2} \) |
| 13 | \( 1 + 0.267T + 13T^{2} \) |
| 17 | \( 1 - 3.97T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 - 8.95T + 31T^{2} \) |
| 37 | \( 1 + 9.45T + 37T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 + 6.10T + 43T^{2} \) |
| 47 | \( 1 + 6.79T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + 3.89T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 - 1.09T + 71T^{2} \) |
| 73 | \( 1 - 6.33T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 7.24T + 83T^{2} \) |
| 89 | \( 1 + 1.60T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.48530464091823355790022343099, −6.86906923104754278348032606603, −6.05195106388394301934912486197, −5.32917625742889445398590348479, −4.73650175926796227677432026103, −3.98014072213205755443416301864, −3.27181407276359772012314884555, −2.04433490800211794748348284066, −1.21997081702224748804272622186, 0,
1.21997081702224748804272622186, 2.04433490800211794748348284066, 3.27181407276359772012314884555, 3.98014072213205755443416301864, 4.73650175926796227677432026103, 5.32917625742889445398590348479, 6.05195106388394301934912486197, 6.86906923104754278348032606603, 7.48530464091823355790022343099
