| L(s) = 1 | − 2·3-s + 4·7-s + 9-s + 5·13-s − 7·19-s − 8·21-s + 3·23-s + 4·27-s − 3·29-s − 5·31-s + 4·37-s − 10·39-s − 12·41-s − 5·43-s + 9·49-s − 6·53-s + 14·57-s − 12·59-s + 10·61-s + 4·63-s + 14·67-s − 6·69-s − 3·71-s + 8·73-s − 4·79-s − 11·81-s + 15·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.38·13-s − 1.60·19-s − 1.74·21-s + 0.625·23-s + 0.769·27-s − 0.557·29-s − 0.898·31-s + 0.657·37-s − 1.60·39-s − 1.87·41-s − 0.762·43-s + 9/7·49-s − 0.824·53-s + 1.85·57-s − 1.56·59-s + 1.28·61-s + 0.503·63-s + 1.71·67-s − 0.722·69-s − 0.356·71-s + 0.936·73-s − 0.450·79-s − 1.22·81-s + 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.91301694637403, −14.39760141578165, −13.73937637971206, −13.23455526339651, −12.70138938669403, −12.15473912993081, −11.49647640031404, −11.20492493985609, −10.84684133517135, −10.52586328473708, −9.692239982789947, −8.881028596091577, −8.457542210452902, −8.117976685035419, −7.356451710784987, −6.588831156648695, −6.295988225601221, −5.623323621992226, −5.021049202473764, −4.739391123784380, −3.928893409684224, −3.366636713804027, −2.201936712843296, −1.657567570846893, −0.9620279804177708, 0,
0.9620279804177708, 1.657567570846893, 2.201936712843296, 3.366636713804027, 3.928893409684224, 4.739391123784380, 5.021049202473764, 5.623323621992226, 6.295988225601221, 6.588831156648695, 7.356451710784987, 8.117976685035419, 8.457542210452902, 8.881028596091577, 9.692239982789947, 10.52586328473708, 10.84684133517135, 11.20492493985609, 11.49647640031404, 12.15473912993081, 12.70138938669403, 13.23455526339651, 13.73937637971206, 14.39760141578165, 14.91301694637403
