| L(s) = 1 | + 5-s − 2·7-s + 6·11-s + 2·17-s + 4·19-s − 23-s + 25-s − 10·29-s − 4·31-s − 2·35-s − 8·37-s − 8·41-s + 4·43-s + 8·47-s − 3·49-s − 6·53-s + 6·55-s − 14·59-s + 10·61-s − 8·71-s − 2·73-s − 12·77-s − 12·79-s + 4·83-s + 2·85-s + 4·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 1.80·11-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.85·29-s − 0.718·31-s − 0.338·35-s − 1.31·37-s − 1.24·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.809·55-s − 1.82·59-s + 1.28·61-s − 0.949·71-s − 0.234·73-s − 1.36·77-s − 1.35·79-s + 0.439·83-s + 0.216·85-s + 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.20882555326816, −15.76124648664647, −14.95673665729504, −14.51732097350571, −13.98784805054961, −13.52436512579191, −12.84798868244471, −12.25715958002399, −11.83618032857646, −11.20452082855752, −10.54455190398793, −9.812005958816177, −9.337477299768821, −9.097069776025600, −8.271465089339811, −7.301864467928180, −7.025945708257690, −6.245809162245020, −5.747326618903393, −5.124579579883202, −4.077888816215101, −3.599683130864953, −2.992404198315491, −1.805040973066070, −1.306524531949232, 0,
1.306524531949232, 1.805040973066070, 2.992404198315491, 3.599683130864953, 4.077888816215101, 5.124579579883202, 5.747326618903393, 6.245809162245020, 7.025945708257690, 7.301864467928180, 8.271465089339811, 9.097069776025600, 9.337477299768821, 9.812005958816177, 10.54455190398793, 11.20452082855752, 11.83618032857646, 12.25715958002399, 12.84798868244471, 13.52436512579191, 13.98784805054961, 14.51732097350571, 14.95673665729504, 15.76124648664647, 16.20882555326816
