| L(s) = 1 | + 5-s − 3·7-s − 3·9-s + 2·11-s + 2·13-s + 7·17-s − 4·19-s − 5·23-s − 4·25-s + 2·29-s + 5·31-s − 3·35-s + 6·37-s + 5·41-s − 10·43-s − 3·45-s + 2·47-s + 2·49-s − 6·53-s + 2·55-s + 6·59-s − 14·61-s + 9·63-s + 2·65-s − 5·67-s − 8·71-s − 73-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 9-s + 0.603·11-s + 0.554·13-s + 1.69·17-s − 0.917·19-s − 1.04·23-s − 4/5·25-s + 0.371·29-s + 0.898·31-s − 0.507·35-s + 0.986·37-s + 0.780·41-s − 1.52·43-s − 0.447·45-s + 0.291·47-s + 2/7·49-s − 0.824·53-s + 0.269·55-s + 0.781·59-s − 1.79·61-s + 1.13·63-s + 0.248·65-s − 0.610·67-s − 0.949·71-s − 0.117·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.61037903535016065477841483001, −6.40220721987787962701609922039, −6.19023729316745707443924489297, −5.71360153889432262066402962990, −4.64615977418933781135435154181, −3.69759192310975933311562942417, −3.18317888675327619718776624075, −2.32975995006181857383286120113, −1.22287579709307756236495037854, 0,
1.22287579709307756236495037854, 2.32975995006181857383286120113, 3.18317888675327619718776624075, 3.69759192310975933311562942417, 4.64615977418933781135435154181, 5.71360153889432262066402962990, 6.19023729316745707443924489297, 6.40220721987787962701609922039, 7.61037903535016065477841483001
