| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 11-s + 15-s − 17-s + 4·19-s − 2·21-s + 23-s + 25-s + 27-s − 3·29-s − 7·31-s + 33-s − 2·35-s + 5·37-s + 11·43-s + 45-s + 3·47-s − 3·49-s − 51-s − 12·53-s + 55-s + 4·57-s + 5·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.258·15-s − 0.242·17-s + 0.917·19-s − 0.436·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s − 1.25·31-s + 0.174·33-s − 0.338·35-s + 0.821·37-s + 1.67·43-s + 0.149·45-s + 0.437·47-s − 3/7·49-s − 0.140·51-s − 1.64·53-s + 0.134·55-s + 0.529·57-s + 0.650·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.83408910326170, −12.40945812873895, −12.06721899862903, −11.27363861617317, −10.96624132137117, −10.58012481007347, −9.797329602433854, −9.555669416509633, −9.230050678093983, −8.900329449725206, −8.158419864659949, −7.688721583366734, −7.321862941774125, −6.756292121613004, −6.337832778758412, −5.742263808199275, −5.436555392462876, −4.668216661335271, −4.226452934534927, −3.573829589376044, −3.201728815077494, −2.665651698066905, −2.083350882397386, −1.482607197770321, −0.8478370840124147, 0,
0.8478370840124147, 1.482607197770321, 2.083350882397386, 2.665651698066905, 3.201728815077494, 3.573829589376044, 4.226452934534927, 4.668216661335271, 5.436555392462876, 5.742263808199275, 6.337832778758412, 6.756292121613004, 7.321862941774125, 7.688721583366734, 8.158419864659949, 8.900329449725206, 9.230050678093983, 9.555669416509633, 9.797329602433854, 10.58012481007347, 10.96624132137117, 11.27363861617317, 12.06721899862903, 12.40945812873895, 12.83408910326170
