| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 3·11-s + 5·13-s + 16-s − 6·17-s − 19-s + 20-s + 3·22-s − 3·23-s + 25-s − 5·26-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 11·37-s + 38-s − 40-s − 3·41-s − 10·43-s − 3·44-s + 3·46-s − 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.904·11-s + 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.229·19-s + 0.223·20-s + 0.639·22-s − 0.625·23-s + 1/5·25-s − 0.980·26-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1.80·37-s + 0.162·38-s − 0.158·40-s − 0.468·41-s − 1.52·43-s − 0.452·44-s + 0.442·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−18.16395441027803, −17.91361660150350, −17.11727182272056, −16.37083490706844, −15.93738164227452, −15.39653496879724, −14.70946386937248, −13.85229174106804, −13.14587523324103, −12.98450448413232, −11.77769008294831, −11.29645844641002, −10.55802549411330, −10.19839182231822, −9.289477817272518, −8.675058805437000, −8.179612029398430, −7.390347999883002, −6.352588588093976, −6.183311392662976, −5.075185265340129, −4.228105526304225, −3.139927008139655, −2.284871326009367, −1.384453095916702, 0,
1.384453095916702, 2.284871326009367, 3.139927008139655, 4.228105526304225, 5.075185265340129, 6.183311392662976, 6.352588588093976, 7.390347999883002, 8.179612029398430, 8.675058805437000, 9.289477817272518, 10.19839182231822, 10.55802549411330, 11.29645844641002, 11.77769008294831, 12.98450448413232, 13.14587523324103, 13.85229174106804, 14.70946386937248, 15.39653496879724, 15.93738164227452, 16.37083490706844, 17.11727182272056, 17.91361660150350, 18.16395441027803
