| L(s) = 1 | − 2.40·3-s − 1.29·5-s − 7-s + 2.79·9-s − 1.09·11-s − 2·13-s + 3.11·15-s + 17-s + 2.40·21-s + 4.81·23-s − 3.32·25-s + 0.483·27-s − 9.32·29-s + 2.97·31-s + 2.63·33-s + 1.29·35-s − 0.907·37-s + 4.81·39-s + 5.90·41-s + 8.31·43-s − 3.62·45-s − 7.64·47-s + 49-s − 2.40·51-s − 5.21·53-s + 1.41·55-s + 2.18·59-s + ⋯ |
| L(s) = 1 | − 1.39·3-s − 0.578·5-s − 0.377·7-s + 0.933·9-s − 0.329·11-s − 0.554·13-s + 0.804·15-s + 0.242·17-s + 0.525·21-s + 1.00·23-s − 0.665·25-s + 0.0930·27-s − 1.73·29-s + 0.533·31-s + 0.458·33-s + 0.218·35-s − 0.149·37-s + 0.771·39-s + 0.921·41-s + 1.26·43-s − 0.539·45-s − 1.11·47-s + 0.142·49-s − 0.337·51-s − 0.715·53-s + 0.190·55-s + 0.284·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.40T + 3T^{2} \) |
| 5 | \( 1 + 1.29T + 5T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 + 9.32T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 + 0.907T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 + 7.64T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 - 2.18T + 59T^{2} \) |
| 61 | \( 1 + 4.40T + 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 7.60T + 97T^{2} \) |
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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
−7.57096799853279096409047740067, −6.69082566983150210893606133028, −6.14771593025336144101653755013, −5.37310499371124220508830894406, −4.91349603746266409927078272020, −4.04807239370015839705486618056, −3.25270958787637503703492646918, −2.18726745538713429309706840397, −0.874295096268019284418240540795, 0,
0.874295096268019284418240540795, 2.18726745538713429309706840397, 3.25270958787637503703492646918, 4.04807239370015839705486618056, 4.91349603746266409927078272020, 5.37310499371124220508830894406, 6.14771593025336144101653755013, 6.69082566983150210893606133028, 7.57096799853279096409047740067
