| L(s) = 1 | − 5-s − 4·11-s + 2·13-s + 2·17-s + 25-s + 6·29-s + 6·37-s − 2·41-s + 12·43-s − 7·49-s + 10·53-s + 4·55-s − 12·59-s + 6·61-s − 2·65-s + 12·67-s − 12·71-s + 2·73-s + 16·79-s − 16·83-s − 2·85-s + 14·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 1/5·25-s + 1.11·29-s + 0.986·37-s − 0.312·41-s + 1.82·43-s − 49-s + 1.37·53-s + 0.539·55-s − 1.56·59-s + 0.768·61-s − 0.248·65-s + 1.46·67-s − 1.42·71-s + 0.234·73-s + 1.80·79-s − 1.75·83-s − 0.216·85-s + 1.48·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.62545251719229, −12.33019832594976, −11.83045950923304, −11.24087268890392, −10.93382251541517, −10.52186361454023, −9.979629858206978, −9.680437002570034, −8.915543112392091, −8.655012612615162, −7.980770540464007, −7.787115658655718, −7.355510919906863, −6.652951659393991, −6.259201630612556, −5.717861059484518, −5.186794433929280, −4.817591769581471, −4.112526129228118, −3.810801952379537, −3.000558421157096, −2.723921269284056, −2.131115110967837, −1.245997159315212, −0.7770912422109666, 0,
0.7770912422109666, 1.245997159315212, 2.131115110967837, 2.723921269284056, 3.000558421157096, 3.810801952379537, 4.112526129228118, 4.817591769581471, 5.186794433929280, 5.717861059484518, 6.259201630612556, 6.652951659393991, 7.355510919906863, 7.787115658655718, 7.980770540464007, 8.655012612615162, 8.915543112392091, 9.680437002570034, 9.979629858206978, 10.52186361454023, 10.93382251541517, 11.24087268890392, 11.83045950923304, 12.33019832594976, 12.62545251719229
