| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 11-s + 2·13-s − 15-s − 3·17-s − 7·19-s + 21-s − 3·23-s + 25-s + 27-s − 3·29-s − 4·31-s + 33-s − 35-s − 10·37-s + 2·39-s − 7·43-s − 45-s + 49-s − 3·51-s + 9·53-s − 55-s − 7·57-s + 3·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.727·17-s − 1.60·19-s + 0.218·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s − 0.718·31-s + 0.174·33-s − 0.169·35-s − 1.64·37-s + 0.320·39-s − 1.06·43-s − 0.149·45-s + 1/7·49-s − 0.420·51-s + 1.23·53-s − 0.134·55-s − 0.927·57-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−8.112704131230471128202873693250, −7.24284676721479087557712081118, −6.65242865352563897437334680458, −5.81283715843067512088619382297, −4.81289466812256589949490912791, −4.04417252527827912594589720431, −3.52766854931708614623244023848, −2.31374469857548530828941086464, −1.59870958895455165459172132126, 0,
1.59870958895455165459172132126, 2.31374469857548530828941086464, 3.52766854931708614623244023848, 4.04417252527827912594589720431, 4.81289466812256589949490912791, 5.81283715843067512088619382297, 6.65242865352563897437334680458, 7.24284676721479087557712081118, 8.112704131230471128202873693250
