| L(s) = 1 | + 2·7-s − 11-s + 2·13-s + 4·17-s − 6·19-s + 8·29-s − 8·31-s − 10·37-s − 8·41-s + 2·43-s − 8·47-s − 3·49-s − 2·53-s − 12·59-s + 10·61-s − 12·67-s − 8·71-s − 6·73-s − 2·77-s − 2·79-s + 16·83-s + 14·89-s + 4·91-s + 2·97-s + 16·101-s − 4·103-s − 10·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s + 1.48·29-s − 1.43·31-s − 1.64·37-s − 1.24·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s − 1.56·59-s + 1.28·61-s − 1.46·67-s − 0.949·71-s − 0.702·73-s − 0.227·77-s − 0.225·79-s + 1.75·83-s + 1.48·89-s + 0.419·91-s + 0.203·97-s + 1.59·101-s − 0.394·103-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.42629306610516342276242914108, −6.57683652024501588774826595788, −6.02836450847810227902800576727, −5.11416709462931679888435399637, −4.72793119489665542925033863841, −3.73568755571893688542885097333, −3.11789050652424218706145111947, −1.99943763380557333346900621518, −1.37472506144710531081245880155, 0,
1.37472506144710531081245880155, 1.99943763380557333346900621518, 3.11789050652424218706145111947, 3.73568755571893688542885097333, 4.72793119489665542925033863841, 5.11416709462931679888435399637, 6.02836450847810227902800576727, 6.57683652024501588774826595788, 7.42629306610516342276242914108
