| L(s) = 1 | − 3-s + 5-s + 9-s − 2·11-s + 13-s − 15-s − 17-s + 8·19-s + 6·23-s + 25-s − 27-s − 4·31-s + 2·33-s − 2·37-s − 39-s + 4·41-s + 4·43-s + 45-s − 7·49-s + 51-s + 6·53-s − 2·55-s − 8·57-s − 4·59-s − 8·61-s + 65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s + 1.83·19-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.718·31-s + 0.348·33-s − 0.328·37-s − 0.160·39-s + 0.624·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.140·51-s + 0.824·53-s − 0.269·55-s − 1.05·57-s − 0.520·59-s − 1.02·61-s + 0.124·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.23369427508486, −12.77357157852029, −12.34809525315493, −11.78390935244893, −11.33336722279176, −10.94451377897944, −10.45630653284713, −10.07779074909433, −9.368885189760035, −9.173893288191170, −8.657459032116271, −7.758913167973088, −7.587912746013064, −7.040652350645319, −6.476923250597091, −5.946558827584537, −5.436045070007276, −5.107252724592921, −4.609622748965748, −3.900682966449501, −3.181725678170361, −2.870959047513282, −2.065749327369794, −1.355090097308406, −0.8833724974268945, 0,
0.8833724974268945, 1.355090097308406, 2.065749327369794, 2.870959047513282, 3.181725678170361, 3.900682966449501, 4.609622748965748, 5.107252724592921, 5.436045070007276, 5.946558827584537, 6.476923250597091, 7.040652350645319, 7.587912746013064, 7.758913167973088, 8.657459032116271, 9.173893288191170, 9.368885189760035, 10.07779074909433, 10.45630653284713, 10.94451377897944, 11.33336722279176, 11.78390935244893, 12.34809525315493, 12.77357157852029, 13.23369427508486
