| L(s) = 1 | − 3-s − 2·7-s + 9-s + 13-s − 17-s + 8·19-s + 2·21-s − 6·23-s − 27-s − 7·29-s + 3·31-s + 2·37-s − 39-s − 12·41-s + 8·43-s + 6·47-s − 3·49-s + 51-s + 9·53-s − 8·57-s + 9·59-s + 2·61-s − 2·63-s − 67-s + 6·69-s + 2·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.242·17-s + 1.83·19-s + 0.436·21-s − 1.25·23-s − 0.192·27-s − 1.29·29-s + 0.538·31-s + 0.328·37-s − 0.160·39-s − 1.87·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.140·51-s + 1.23·53-s − 1.05·57-s + 1.17·59-s + 0.256·61-s − 0.251·63-s − 0.122·67-s + 0.722·69-s + 0.237·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−13.61259412519860, −13.19815312026219, −12.82311595885221, −12.03860910081362, −11.89406415784689, −11.40377466073362, −10.85807465013701, −10.21711861184310, −9.868672345995271, −9.517282468807377, −8.894136206849069, −8.338175230366546, −7.724791537658227, −7.169307522993551, −6.882381140924907, −6.117869308169805, −5.724810257485655, −5.366050683208354, −4.650770101985016, −3.931383607874505, −3.607472885957077, −2.904910452040025, −2.232997659868413, −1.466285053026025, −0.7634810112001259, 0,
0.7634810112001259, 1.466285053026025, 2.232997659868413, 2.904910452040025, 3.607472885957077, 3.931383607874505, 4.650770101985016, 5.366050683208354, 5.724810257485655, 6.117869308169805, 6.882381140924907, 7.169307522993551, 7.724791537658227, 8.338175230366546, 8.894136206849069, 9.517282468807377, 9.868672345995271, 10.21711861184310, 10.85807465013701, 11.40377466073362, 11.89406415784689, 12.03860910081362, 12.82311595885221, 13.19815312026219, 13.61259412519860
