| L(s) = 1 | − 3-s + 7-s + 9-s − 4·13-s + 3·17-s + 7·19-s − 21-s + 9·23-s − 27-s + 3·29-s + 2·31-s + 4·37-s + 4·39-s + 6·41-s − 43-s + 6·47-s + 49-s − 3·51-s − 3·53-s − 7·57-s − 9·59-s + 61-s + 63-s + 10·67-s − 9·69-s + 6·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.727·17-s + 1.60·19-s − 0.218·21-s + 1.87·23-s − 0.192·27-s + 0.557·29-s + 0.359·31-s + 0.657·37-s + 0.640·39-s + 0.937·41-s − 0.152·43-s + 0.875·47-s + 1/7·49-s − 0.420·51-s − 0.412·53-s − 0.927·57-s − 1.17·59-s + 0.128·61-s + 0.125·63-s + 1.22·67-s − 1.08·69-s + 0.712·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.169559374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.169559374\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.56725626967760, −12.43970073740724, −11.92917970075143, −11.46408253729448, −10.97198658682012, −10.71787806527695, −9.951271282538547, −9.608124646473379, −9.333873560189977, −8.667936903618593, −7.918522565148488, −7.724738680574934, −7.104393853483781, −6.828426920977414, −6.139567508835220, −5.449195847166952, −5.252086813177958, −4.752479392908315, −4.262600500995758, −3.492562805268433, −2.904209553002359, −2.544052930323564, −1.632684134723667, −0.9568438332834628, −0.6350794474326701,
0.6350794474326701, 0.9568438332834628, 1.632684134723667, 2.544052930323564, 2.904209553002359, 3.492562805268433, 4.262600500995758, 4.752479392908315, 5.252086813177958, 5.449195847166952, 6.139567508835220, 6.828426920977414, 7.104393853483781, 7.724738680574934, 7.918522565148488, 8.667936903618593, 9.333873560189977, 9.608124646473379, 9.951271282538547, 10.71787806527695, 10.97198658682012, 11.46408253729448, 11.92917970075143, 12.43970073740724, 12.56725626967760
