| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 7-s − 2·9-s + 2·10-s + 2·12-s − 5·13-s − 2·14-s + 15-s − 4·16-s + 2·17-s − 4·18-s − 2·19-s + 2·20-s − 21-s + 2·23-s + 25-s − 10·26-s − 5·27-s − 2·28-s − 6·29-s + 2·30-s − 4·31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s − 2/3·9-s + 0.632·10-s + 0.577·12-s − 1.38·13-s − 0.534·14-s + 0.258·15-s − 16-s + 0.485·17-s − 0.942·18-s − 0.458·19-s + 0.447·20-s − 0.218·21-s + 0.417·23-s + 1/5·25-s − 1.96·26-s − 0.962·27-s − 0.377·28-s − 1.11·29-s + 0.365·30-s − 0.718·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.150190385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150190385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 53 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.85259384531873, −13.22041484284277, −12.98564358897737, −12.48706372167612, −11.96745741825057, −11.54090058846452, −11.01524675979770, −10.30681970358539, −9.781224824976591, −9.355627153300983, −8.801041989009631, −8.358957509342496, −7.506983110631054, −7.181906618245031, −6.476823364589000, −6.063172480343727, −5.359634277349521, −5.102351507625643, −4.566478085471603, −3.703384235543029, −3.247206004102388, −2.979821717478618, −1.998638347728380, −1.922614314523426, −0.3069395950909021,
0.3069395950909021, 1.922614314523426, 1.998638347728380, 2.979821717478618, 3.247206004102388, 3.703384235543029, 4.566478085471603, 5.102351507625643, 5.359634277349521, 6.063172480343727, 6.476823364589000, 7.181906618245031, 7.506983110631054, 8.358957509342496, 8.801041989009631, 9.355627153300983, 9.781224824976591, 10.30681970358539, 11.01524675979770, 11.54090058846452, 11.96745741825057, 12.48706372167612, 12.98564358897737, 13.22041484284277, 13.85259384531873
