| L(s) = 1 | + 2·3-s − 5-s − 7-s + 9-s + 11-s + 2·13-s − 2·15-s − 6·17-s − 2·19-s − 2·21-s + 6·23-s + 25-s − 4·27-s − 8·31-s + 2·33-s + 35-s − 4·37-s + 4·39-s + 12·41-s + 4·43-s − 45-s − 12·47-s + 49-s − 12·51-s − 55-s − 4·57-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.458·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.43·31-s + 0.348·33-s + 0.169·35-s − 0.657·37-s + 0.640·39-s + 1.87·41-s + 0.609·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 1.68·51-s − 0.134·55-s − 0.529·57-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.73312059354095163019771183523, −7.13525926896862965822695992878, −6.44943906930903131644251512371, −5.61585292738124780236283596198, −4.52724348464294377184119647792, −3.92433729351186389637482846461, −3.17803512612532361110671420339, −2.49490722893490036729934196888, −1.51315078543919455507762455132, 0,
1.51315078543919455507762455132, 2.49490722893490036729934196888, 3.17803512612532361110671420339, 3.92433729351186389637482846461, 4.52724348464294377184119647792, 5.61585292738124780236283596198, 6.44943906930903131644251512371, 7.13525926896862965822695992878, 7.73312059354095163019771183523
