| L(s) = 1 | − 2·3-s + 9-s − 2·11-s + 6·13-s − 17-s − 4·19-s + 4·23-s − 5·25-s + 4·27-s − 8·31-s + 4·33-s + 4·37-s − 12·39-s + 6·41-s − 8·43-s − 8·47-s − 7·49-s + 2·51-s − 10·53-s + 8·57-s − 12·61-s − 8·67-s − 8·69-s + 12·71-s + 2·73-s + 10·75-s − 4·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.603·11-s + 1.66·13-s − 0.242·17-s − 0.917·19-s + 0.834·23-s − 25-s + 0.769·27-s − 1.43·31-s + 0.696·33-s + 0.657·37-s − 1.92·39-s + 0.937·41-s − 1.21·43-s − 1.16·47-s − 49-s + 0.280·51-s − 1.37·53-s + 1.05·57-s − 1.53·61-s − 0.977·67-s − 0.963·69-s + 1.42·71-s + 0.234·73-s + 1.15·75-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.523789490321376043521603832717, −8.601865042316678712027569971509, −7.82524899527661099789413810744, −6.61559225862052910410642757858, −6.07731647450232436470441656848, −5.29038885848037918717721857208, −4.32353360786394336037815909787, −3.17822913513543322234454785908, −1.58461391583759898942398456470, 0,
1.58461391583759898942398456470, 3.17822913513543322234454785908, 4.32353360786394336037815909787, 5.29038885848037918717721857208, 6.07731647450232436470441656848, 6.61559225862052910410642757858, 7.82524899527661099789413810744, 8.601865042316678712027569971509, 9.523789490321376043521603832717
