| L(s) = 1 | − 7-s − 3·9-s − 5·11-s − 5·13-s − 6·17-s − 5·19-s + 3·29-s + 4·31-s − 6·37-s − 9·41-s + 5·43-s − 10·47-s − 6·49-s − 12·53-s − 2·59-s + 10·61-s + 3·63-s + 4·67-s − 6·71-s + 15·73-s + 5·77-s − 15·79-s + 9·81-s − 7·83-s + 10·89-s + 5·91-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.50·11-s − 1.38·13-s − 1.45·17-s − 1.14·19-s + 0.557·29-s + 0.718·31-s − 0.986·37-s − 1.40·41-s + 0.762·43-s − 1.45·47-s − 6/7·49-s − 1.64·53-s − 0.260·59-s + 1.28·61-s + 0.377·63-s + 0.488·67-s − 0.712·71-s + 1.75·73-s + 0.569·77-s − 1.68·79-s + 81-s − 0.768·83-s + 1.05·89-s + 0.524·91-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.20743165390675, −12.76666353670297, −12.42149132188081, −11.86944983051091, −11.23373765550092, −11.01229967010054, −10.40297359976151, −9.962216225209009, −9.641869440099252, −8.834470196512242, −8.534524784953805, −8.084879611488656, −7.648459765520313, −6.851050349243827, −6.614952490237457, −6.098186485662438, −5.365183268674080, −4.874630468628633, −4.724358466661874, −3.876790646030350, −3.093571558391978, −2.732435491183429, −2.265736090756400, −1.723966170734502, −0.3680168119170131, 0,
0.3680168119170131, 1.723966170734502, 2.265736090756400, 2.732435491183429, 3.093571558391978, 3.876790646030350, 4.724358466661874, 4.874630468628633, 5.365183268674080, 6.098186485662438, 6.614952490237457, 6.851050349243827, 7.648459765520313, 8.084879611488656, 8.534524784953805, 8.834470196512242, 9.641869440099252, 9.962216225209009, 10.40297359976151, 11.01229967010054, 11.23373765550092, 11.86944983051091, 12.42149132188081, 12.76666353670297, 13.20743165390675
