| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 11-s − 13-s + 16-s + 3·19-s − 20-s + 22-s − 7·23-s + 25-s − 26-s + 8·29-s + 2·31-s + 32-s + 11·37-s + 3·38-s − 40-s − 11·41-s + 8·43-s + 44-s − 7·46-s − 5·47-s + 50-s − 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.277·13-s + 1/4·16-s + 0.688·19-s − 0.223·20-s + 0.213·22-s − 1.45·23-s + 1/5·25-s − 0.196·26-s + 1.48·29-s + 0.359·31-s + 0.176·32-s + 1.80·37-s + 0.486·38-s − 0.158·40-s − 1.71·41-s + 1.21·43-s + 0.150·44-s − 1.03·46-s − 0.729·47-s + 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.868370817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.868370817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.150149311642982143720753419665, −7.64520969807885352996375891508, −6.76465999798410197970364155905, −6.17955291083753433044886873040, −5.33029838161233124836993267190, −4.53259138208873278855056003349, −3.90593184920529201683720597533, −3.03073886106815598508910759478, −2.15666973839873817490152762248, −0.864051942346481283386568118080,
0.864051942346481283386568118080, 2.15666973839873817490152762248, 3.03073886106815598508910759478, 3.90593184920529201683720597533, 4.53259138208873278855056003349, 5.33029838161233124836993267190, 6.17955291083753433044886873040, 6.76465999798410197970364155905, 7.64520969807885352996375891508, 8.150149311642982143720753419665
