| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 11-s + 6·13-s + 16-s − 4·19-s + 20-s − 22-s + 2·23-s + 25-s − 6·26-s + 10·29-s − 4·31-s − 32-s − 10·37-s + 4·38-s − 40-s − 6·41-s + 8·43-s + 44-s − 2·46-s − 4·47-s − 7·49-s − 50-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 + 1.66·13-s + 1/4·16-s − 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.417·23-s + 1/5·25-s − 1.17·26-s + 1.85·29-s − 0.718·31-s − 0.176·32-s − 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s + 1.21·43-s + 0.150·44-s − 0.294·46-s − 0.583·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.99287795428550, −12.43613651325235, −11.96955184557342, −11.55089898778863, −10.90303525334774, −10.54138333885134, −10.44912162740327, −9.656031934844454, −9.194988946365941, −8.798222184343165, −8.352326299683733, −8.135849587800457, −7.317017882646484, −6.769457835526161, −6.417000627254608, −6.143189697770539, −5.370278127819510, −4.994400158624569, −4.240789624994642, −3.621618662638310, −3.299051287886191, −2.478628915226612, −1.987844939386572, −1.312748650022390, −0.9215138727297576, 0,
0.9215138727297576, 1.312748650022390, 1.987844939386572, 2.478628915226612, 3.299051287886191, 3.621618662638310, 4.240789624994642, 4.994400158624569, 5.370278127819510, 6.143189697770539, 6.417000627254608, 6.769457835526161, 7.317017882646484, 8.135849587800457, 8.352326299683733, 8.798222184343165, 9.194988946365941, 9.656031934844454, 10.44912162740327, 10.54138333885134, 10.90303525334774, 11.55089898778863, 11.96955184557342, 12.43613651325235, 12.99287795428550
