| L(s) = 1 | − 5-s − 3·7-s − 5·13-s − 5·17-s + 19-s + 4·23-s + 25-s + 5·29-s + 2·31-s + 3·35-s + 3·37-s − 2·41-s + 2·43-s − 6·47-s + 2·49-s − 4·59-s + 8·61-s + 5·65-s + 6·67-s − 3·71-s − 10·73-s − 10·79-s − 9·83-s + 5·85-s + 12·89-s + 15·91-s − 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 1.38·13-s − 1.21·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.928·29-s + 0.359·31-s + 0.507·35-s + 0.493·37-s − 0.312·41-s + 0.304·43-s − 0.875·47-s + 2/7·49-s − 0.520·59-s + 1.02·61-s + 0.620·65-s + 0.733·67-s − 0.356·71-s − 1.17·73-s − 1.12·79-s − 0.987·83-s + 0.542·85-s + 1.27·89-s + 1.57·91-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.90847172756471, −14.56444699960074, −13.86972633288954, −13.26938009163254, −12.83780520772389, −12.49822159378724, −11.75318127879707, −11.47199729544972, −10.74199940333770, −10.09782357510672, −9.800859353663183, −9.132365266012536, −8.716420436321793, −8.002546913818347, −7.382019521001547, −6.819035892827922, −6.543948455408471, −5.770242618966979, −4.968834409821732, −4.560503076365095, −3.904225942627500, −2.963882499989205, −2.807866462456840, −1.883260371876704, −0.7369912554908880, 0,
0.7369912554908880, 1.883260371876704, 2.807866462456840, 2.963882499989205, 3.904225942627500, 4.560503076365095, 4.968834409821732, 5.770242618966979, 6.543948455408471, 6.819035892827922, 7.382019521001547, 8.002546913818347, 8.716420436321793, 9.132365266012536, 9.800859353663183, 10.09782357510672, 10.74199940333770, 11.47199729544972, 11.75318127879707, 12.49822159378724, 12.83780520772389, 13.26938009163254, 13.86972633288954, 14.56444699960074, 14.90847172756471
