| L(s) = 1 | + 3-s + 5-s + 9-s + 2·11-s − 4·13-s + 15-s − 6·17-s − 6·19-s − 8·23-s + 25-s + 27-s − 2·29-s − 10·31-s + 2·33-s + 2·37-s − 4·39-s − 10·41-s − 4·43-s + 45-s + 8·47-s − 6·51-s + 4·53-s + 2·55-s − 6·57-s + 8·59-s − 6·61-s − 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s − 1.37·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.79·31-s + 0.348·33-s + 0.328·37-s − 0.640·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s − 0.840·51-s + 0.549·53-s + 0.269·55-s − 0.794·57-s + 1.04·59-s − 0.768·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.576753063635997780354724816607, −7.61991010314532395293141674162, −6.85500220249878455819000825708, −6.24416431678863774019927421815, −5.23406477936034179422601656737, −4.31129161226649985989005461361, −3.67509976173477623917923137546, −2.25371126242634563634491015097, −1.97163031330732203509706193529, 0,
1.97163031330732203509706193529, 2.25371126242634563634491015097, 3.67509976173477623917923137546, 4.31129161226649985989005461361, 5.23406477936034179422601656737, 6.24416431678863774019927421815, 6.85500220249878455819000825708, 7.61991010314532395293141674162, 8.576753063635997780354724816607
