L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 6·11-s + 2·13-s + 4·15-s + 4·17-s + 4·19-s − 8·23-s − 25-s + 4·27-s + 6·29-s − 12·33-s − 2·37-s − 4·39-s − 6·41-s − 6·43-s − 2·45-s − 12·47-s − 8·51-s − 2·53-s − 12·55-s − 8·57-s + 12·59-s − 2·61-s − 4·65-s + 4·67-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 1.03·15-s + 0.970·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s − 2.08·33-s − 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s − 0.298·45-s − 1.75·47-s − 1.12·51-s − 0.274·53-s − 1.61·55-s − 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.199239286\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.199239286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.25682033215477, −11.89679910425620, −11.58368896860407, −11.46741548263353, −10.83506147442864, −10.20749475360323, −9.782607309850116, −9.585717360385915, −8.594535881861736, −8.456501947273202, −7.957588682841908, −7.365384856158280, −6.807040185027274, −6.356134550268175, −6.184772959681038, −5.462606802804672, −5.059830500905141, −4.511505072568130, −3.905966246811685, −3.517794445116528, −3.228093372423378, −2.145974939118334, −1.444477789062571, −1.018705517462766, −0.3659826060374281,
0.3659826060374281, 1.018705517462766, 1.444477789062571, 2.145974939118334, 3.228093372423378, 3.517794445116528, 3.905966246811685, 4.511505072568130, 5.059830500905141, 5.462606802804672, 6.184772959681038, 6.356134550268175, 6.807040185027274, 7.365384856158280, 7.957588682841908, 8.456501947273202, 8.594535881861736, 9.585717360385915, 9.782607309850116, 10.20749475360323, 10.83506147442864, 11.46741548263353, 11.58368896860407, 11.89679910425620, 12.25682033215477