| L(s) = 1 | − 2·3-s − 7-s + 9-s − 5·11-s − 13-s − 4·17-s + 7·19-s + 2·21-s + 4·27-s + 5·29-s − 2·31-s + 10·33-s − 2·37-s + 2·39-s + 11·41-s − 43-s − 8·47-s − 6·49-s + 8·51-s − 14·57-s + 14·59-s − 10·61-s − 63-s + 8·67-s + 10·71-s − 7·73-s + 5·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.970·17-s + 1.60·19-s + 0.436·21-s + 0.769·27-s + 0.928·29-s − 0.359·31-s + 1.74·33-s − 0.328·37-s + 0.320·39-s + 1.71·41-s − 0.152·43-s − 1.16·47-s − 6/7·49-s + 1.12·51-s − 1.85·57-s + 1.82·59-s − 1.28·61-s − 0.125·63-s + 0.977·67-s + 1.18·71-s − 0.819·73-s + 0.569·77-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 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.11538442127876, −12.73657294020162, −12.31791003084574, −11.76179971896298, −11.38135404254818, −10.90250575605639, −10.61114596753919, −9.958834750265412, −9.694391105751764, −9.090456697956630, −8.448145240583878, −7.961447396305795, −7.467830124560412, −6.977248806271893, −6.401257917369032, −6.058420603862197, −5.348807361508426, −5.071235056657355, −4.764938608390026, −3.932543113164586, −3.264868881770361, −2.684983351416005, −2.273615223380748, −1.273405983393170, −0.6001365689020406, 0,
0.6001365689020406, 1.273405983393170, 2.273615223380748, 2.684983351416005, 3.264868881770361, 3.932543113164586, 4.764938608390026, 5.071235056657355, 5.348807361508426, 6.058420603862197, 6.401257917369032, 6.977248806271893, 7.467830124560412, 7.961447396305795, 8.448145240583878, 9.090456697956630, 9.694391105751764, 9.958834750265412, 10.61114596753919, 10.90250575605639, 11.38135404254818, 11.76179971896298, 12.31791003084574, 12.73657294020162, 13.11538442127876
