| L(s) = 1 | + 3-s + 7-s + 9-s − 4·11-s − 4·13-s − 6·17-s − 6·19-s + 21-s − 23-s + 27-s + 6·29-s − 2·31-s − 4·33-s + 2·37-s − 4·39-s + 2·41-s + 43-s + 47-s − 6·49-s − 6·51-s − 6·53-s − 6·57-s − 8·59-s − 10·61-s + 63-s − 7·67-s − 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 1.37·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.696·33-s + 0.328·37-s − 0.640·39-s + 0.312·41-s + 0.152·43-s + 0.145·47-s − 6/7·49-s − 0.840·51-s − 0.824·53-s − 0.794·57-s − 1.04·59-s − 1.28·61-s + 0.125·63-s − 0.855·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9634627714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9634627714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.51877709104223, −14.04669095921363, −13.33035189081478, −13.11934042400439, −12.43095191587528, −12.16074456765963, −11.21829741341916, −10.86287634582067, −10.38348708626257, −9.862472671687409, −9.150353597592079, −8.806152701415371, −8.005270608551197, −7.891201703811737, −7.153472176008569, −6.536726348413837, −6.044658190704569, −5.135240583767356, −4.564684382548106, −4.411766933576322, −3.362827509366773, −2.607988030487885, −2.300167586594270, −1.605345439043598, −0.3061212897846791,
0.3061212897846791, 1.605345439043598, 2.300167586594270, 2.607988030487885, 3.362827509366773, 4.411766933576322, 4.564684382548106, 5.135240583767356, 6.044658190704569, 6.536726348413837, 7.153472176008569, 7.891201703811737, 8.005270608551197, 8.806152701415371, 9.150353597592079, 9.862472671687409, 10.38348708626257, 10.86287634582067, 11.21829741341916, 12.16074456765963, 12.43095191587528, 13.11934042400439, 13.33035189081478, 14.04669095921363, 14.51877709104223
