| L(s) = 1 | − 5-s − 4·11-s + 13-s − 2·17-s − 4·23-s + 25-s + 10·29-s + 4·31-s + 6·37-s − 2·41-s − 4·43-s + 14·53-s + 4·55-s + 4·59-s + 10·61-s − 65-s − 4·67-s + 2·73-s + 12·83-s + 2·85-s − 2·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.92·53-s + 0.539·55-s + 0.520·59-s + 1.28·61-s − 0.124·65-s − 0.488·67-s + 0.234·73-s + 1.31·83-s + 0.216·85-s − 0.211·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.193439473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.193439473\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.97670312999805, −12.48944228553285, −11.79355819512789, −11.75571463085393, −11.09591311858076, −10.45649380166687, −10.24550907584072, −9.857997030082414, −9.081380618343471, −8.602321997722958, −8.112611890194291, −7.964164826806378, −7.208619479572860, −6.770596199767648, −6.258171305008221, −5.716751591930082, −5.148990179244318, −4.631668447309145, −4.218932338460882, −3.574308018279737, −2.956258042465454, −2.450005885031661, −1.964387898393438, −0.9199351399347756, −0.5011539515759570,
0.5011539515759570, 0.9199351399347756, 1.964387898393438, 2.450005885031661, 2.956258042465454, 3.574308018279737, 4.218932338460882, 4.631668447309145, 5.148990179244318, 5.716751591930082, 6.258171305008221, 6.770596199767648, 7.208619479572860, 7.964164826806378, 8.112611890194291, 8.602321997722958, 9.081380618343471, 9.857997030082414, 10.24550907584072, 10.45649380166687, 11.09591311858076, 11.75571463085393, 11.79355819512789, 12.48944228553285, 12.97670312999805
