| L(s) = 1 | − 2-s + 4-s − 8-s + 2·11-s + 13-s + 16-s + 3·17-s − 7·19-s − 2·22-s + 8·23-s − 26-s + 6·29-s + 4·31-s − 32-s − 3·34-s − 12·37-s + 7·38-s + 7·41-s + 9·43-s + 2·44-s − 8·46-s + 2·47-s + 52-s − 4·53-s − 6·58-s − 12·59-s − 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.603·11-s + 0.277·13-s + 1/4·16-s + 0.727·17-s − 1.60·19-s − 0.426·22-s + 1.66·23-s − 0.196·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 1.97·37-s + 1.13·38-s + 1.09·41-s + 1.37·43-s + 0.301·44-s − 1.17·46-s + 0.291·47-s + 0.138·52-s − 0.549·53-s − 0.787·58-s − 1.56·59-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.182917469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.182917469\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−12.64778808181457, −12.22475918157323, −11.87654131628543, −11.10065499771646, −10.88552927439979, −10.45950072003706, −10.05751947065751, −9.306135524960715, −9.008110804170915, −8.755422626626747, −7.996157074216765, −7.816192618475433, −7.064848402605706, −6.598436404353042, −6.358907760583616, −5.753698882586244, −5.044364206830368, −4.668401353186537, −3.942705896959557, −3.494425666841203, −2.803588040034735, −2.373931571452039, −1.597453925425756, −1.065624022078970, −0.5044920620475667,
0.5044920620475667, 1.065624022078970, 1.597453925425756, 2.373931571452039, 2.803588040034735, 3.494425666841203, 3.942705896959557, 4.668401353186537, 5.044364206830368, 5.753698882586244, 6.358907760583616, 6.598436404353042, 7.064848402605706, 7.816192618475433, 7.996157074216765, 8.755422626626747, 9.008110804170915, 9.306135524960715, 10.05751947065751, 10.45950072003706, 10.88552927439979, 11.10065499771646, 11.87654131628543, 12.22475918157323, 12.64778808181457
