| L(s) = 1 | + 3·5-s + 2·11-s − 17-s + 3·19-s − 5·23-s + 4·25-s − 6·29-s + 3·37-s + 43-s − 6·47-s + 6·53-s + 6·55-s + 5·59-s + 10·61-s + 7·67-s + 3·71-s − 4·73-s − 16·79-s − 12·83-s − 3·85-s − 5·89-s + 9·95-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.603·11-s − 0.242·17-s + 0.688·19-s − 1.04·23-s + 4/5·25-s − 1.11·29-s + 0.493·37-s + 0.152·43-s − 0.875·47-s + 0.824·53-s + 0.809·55-s + 0.650·59-s + 1.28·61-s + 0.855·67-s + 0.356·71-s − 0.468·73-s − 1.80·79-s − 1.31·83-s − 0.325·85-s − 0.529·89-s + 0.923·95-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.86700328355701, −13.30041208938674, −12.96119532001184, −12.49836680484438, −11.72953896381809, −11.46486316964324, −10.95830460777491, −10.11567971820216, −9.939355401277149, −9.563853398033641, −8.956184243909945, −8.532523875018140, −7.913918544573987, −7.219246994272780, −6.858545654504540, −6.141944636047955, −5.835053942435781, −5.360362615557877, −4.759290856078913, −3.981969563433301, −3.615909938970202, −2.697573872043626, −2.240170514736555, −1.614462115851535, −1.061657245140739, 0,
1.061657245140739, 1.614462115851535, 2.240170514736555, 2.697573872043626, 3.615909938970202, 3.981969563433301, 4.759290856078913, 5.360362615557877, 5.835053942435781, 6.141944636047955, 6.858545654504540, 7.219246994272780, 7.913918544573987, 8.532523875018140, 8.956184243909945, 9.563853398033641, 9.939355401277149, 10.11567971820216, 10.95830460777491, 11.46486316964324, 11.72953896381809, 12.49836680484438, 12.96119532001184, 13.30041208938674, 13.86700328355701
