| L(s) = 1 | − 5-s − 3·9-s + 4·11-s − 13-s − 6·17-s − 4·19-s + 25-s − 2·29-s + 4·31-s − 6·37-s − 6·41-s − 8·43-s + 3·45-s − 7·49-s + 2·53-s − 4·55-s − 4·59-s − 10·61-s + 65-s − 12·67-s + 4·71-s + 14·73-s + 16·79-s + 9·81-s − 12·83-s + 6·85-s + 2·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s + 1.20·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.447·45-s − 49-s + 0.274·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.124·65-s − 1.46·67-s + 0.474·71-s + 1.63·73-s + 1.80·79-s + 81-s − 1.31·83-s + 0.650·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.289516882563505753717844836319, −8.727729757648928794693246501463, −8.020675637712706628115157809788, −6.75431098097545836131130362927, −6.35751715549737235991528287917, −5.05992604483688855087742886696, −4.17780986294689963383507951066, −3.17194419077005105265156458280, −1.89545288791996576433742409664, 0,
1.89545288791996576433742409664, 3.17194419077005105265156458280, 4.17780986294689963383507951066, 5.05992604483688855087742886696, 6.35751715549737235991528287917, 6.75431098097545836131130362927, 8.020675637712706628115157809788, 8.727729757648928794693246501463, 9.289516882563505753717844836319
