L(s) = 1 | + 5-s − 4·7-s − 4·13-s + 4·19-s − 8·23-s + 25-s − 4·29-s − 4·31-s − 4·35-s − 2·37-s + 4·41-s + 12·43-s − 8·47-s + 9·49-s + 10·53-s − 4·65-s + 4·67-s + 4·71-s + 4·73-s − 12·79-s − 16·83-s + 6·89-s + 16·91-s + 4·95-s − 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 1.10·13-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.742·29-s − 0.718·31-s − 0.676·35-s − 0.328·37-s + 0.624·41-s + 1.82·43-s − 1.16·47-s + 9/7·49-s + 1.37·53-s − 0.496·65-s + 0.488·67-s + 0.474·71-s + 0.468·73-s − 1.35·79-s − 1.75·83-s + 0.635·89-s + 1.67·91-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7380575314\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7380575314\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.99263075946505, −13.31595707495196, −12.85376235149211, −12.56705290553046, −12.00029731017844, −11.54378328729616, −10.86570164557083, −10.10993356055010, −9.999022788239830, −9.440677314836233, −9.140872346207686, −8.421802965523278, −7.617760624830063, −7.331436144903252, −6.784621999393206, −6.093553827655698, −5.759537499712075, −5.250814505153204, −4.451736432670748, −3.785814473630532, −3.368885443427463, −2.514952892390691, −2.257650733308267, −1.251316685925353, −0.2774765693331784,
0.2774765693331784, 1.251316685925353, 2.257650733308267, 2.514952892390691, 3.368885443427463, 3.785814473630532, 4.451736432670748, 5.250814505153204, 5.759537499712075, 6.093553827655698, 6.784621999393206, 7.331436144903252, 7.617760624830063, 8.421802965523278, 9.140872346207686, 9.440677314836233, 9.999022788239830, 10.10993356055010, 10.86570164557083, 11.54378328729616, 12.00029731017844, 12.56705290553046, 12.85376235149211, 13.31595707495196, 13.99263075946505