L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 2·11-s + 2·13-s + 15-s + 17-s + 4·19-s + 21-s + 2·23-s + 25-s + 27-s − 2·29-s − 8·31-s − 2·33-s + 35-s − 6·37-s + 2·39-s − 10·41-s + 45-s + 2·47-s + 49-s + 51-s − 10·53-s − 2·55-s + 4·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s + 0.917·19-s + 0.218·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.348·33-s + 0.169·35-s − 0.986·37-s + 0.320·39-s − 1.56·41-s + 0.149·45-s + 0.291·47-s + 1/7·49-s + 0.140·51-s − 1.37·53-s − 0.269·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.00164217796221, −13.39388958210014, −12.90183009960524, −12.64411217362367, −11.89086021662993, −11.37113046695458, −10.95297667595034, −10.38157718595778, −9.947028216255689, −9.394316981267881, −8.990899159783293, −8.368650584707933, −8.039598010908123, −7.416936481212947, −6.924599582940574, −6.463400614383326, −5.532981242361434, −5.311254679252819, −4.866375956001896, −3.826596999819584, −3.602221815902412, −2.901011061349602, −2.242339993369095, −1.645469848472941, −1.073156815192800, 0,
1.073156815192800, 1.645469848472941, 2.242339993369095, 2.901011061349602, 3.602221815902412, 3.826596999819584, 4.866375956001896, 5.311254679252819, 5.532981242361434, 6.463400614383326, 6.924599582940574, 7.416936481212947, 8.039598010908123, 8.368650584707933, 8.990899159783293, 9.394316981267881, 9.947028216255689, 10.38157718595778, 10.95297667595034, 11.37113046695458, 11.89086021662993, 12.64411217362367, 12.90183009960524, 13.39388958210014, 14.00164217796221