L(s) = 1 | − 5-s − 2·11-s + 13-s − 17-s − 4·19-s + 8·23-s − 4·25-s − 3·29-s + 3·31-s + 4·37-s − 41-s + 4·43-s − 9·47-s + 2·55-s + 11·59-s − 65-s + 12·67-s + 4·71-s + 6·73-s + 9·83-s + 85-s + 6·89-s + 4·95-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.603·11-s + 0.277·13-s − 0.242·17-s − 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.557·29-s + 0.538·31-s + 0.657·37-s − 0.156·41-s + 0.609·43-s − 1.31·47-s + 0.269·55-s + 1.43·59-s − 0.124·65-s + 1.46·67-s + 0.474·71-s + 0.702·73-s + 0.987·83-s + 0.108·85-s + 0.635·89-s + 0.410·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.60090803328191, −13.24674170013193, −12.99527322904772, −12.36241595377429, −11.89154467925374, −11.24435736122943, −10.97283936326020, −10.59165652429476, −9.805862120687622, −9.489981108354871, −8.861459595465693, −8.304419074075001, −7.966528057300578, −7.448885942975876, −6.638282182450475, −6.578407867338779, −5.695565265983388, −5.147506114343040, −4.760046693050552, −3.857986606322921, −3.764500643737669, −2.734906755208392, −2.439128383384931, −1.559884385288574, −0.7831449537514705, 0,
0.7831449537514705, 1.559884385288574, 2.439128383384931, 2.734906755208392, 3.764500643737669, 3.857986606322921, 4.760046693050552, 5.147506114343040, 5.695565265983388, 6.578407867338779, 6.638282182450475, 7.448885942975876, 7.966528057300578, 8.304419074075001, 8.861459595465693, 9.489981108354871, 9.805862120687622, 10.59165652429476, 10.97283936326020, 11.24435736122943, 11.89154467925374, 12.36241595377429, 12.99527322904772, 13.24674170013193, 13.60090803328191