| L(s) = 1 | − 2·2-s + 3·3-s + 2·4-s + 4·5-s − 6·6-s + 6·9-s − 8·10-s − 6·11-s + 6·12-s + 4·13-s + 12·15-s − 4·16-s + 4·17-s − 12·18-s + 7·19-s + 8·20-s + 12·22-s − 6·23-s + 11·25-s − 8·26-s + 9·27-s − 6·29-s − 24·30-s + 2·31-s + 8·32-s − 18·33-s − 8·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 4-s + 1.78·5-s − 2.44·6-s + 2·9-s − 2.52·10-s − 1.80·11-s + 1.73·12-s + 1.10·13-s + 3.09·15-s − 16-s + 0.970·17-s − 2.82·18-s + 1.60·19-s + 1.78·20-s + 2.55·22-s − 1.25·23-s + 11/5·25-s − 1.56·26-s + 1.73·27-s − 1.11·29-s − 4.38·30-s + 0.359·31-s + 1.41·32-s − 3.13·33-s − 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248773 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248773 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.476757167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.476757167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 5077 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.21495713217107, −12.60174210332792, −11.91651749001402, −11.11899785196956, −10.64597468049235, −10.13797064707568, −9.986775786905225, −9.496743052984339, −9.309489571212765, −8.626136754440598, −8.246639038746404, −7.964867690579693, −7.386016565777356, −7.124755930679463, −6.256111629003216, −5.734638586791581, −5.312800747384428, −4.710552736043915, −3.788673222306224, −3.252611562616229, −2.724731424248563, −2.274584189815714, −1.724627432408949, −1.387773402928849, −0.6364604858554898,
0.6364604858554898, 1.387773402928849, 1.724627432408949, 2.274584189815714, 2.724731424248563, 3.252611562616229, 3.788673222306224, 4.710552736043915, 5.312800747384428, 5.734638586791581, 6.256111629003216, 7.124755930679463, 7.386016565777356, 7.964867690579693, 8.246639038746404, 8.626136754440598, 9.309489571212765, 9.496743052984339, 9.986775786905225, 10.13797064707568, 10.64597468049235, 11.11899785196956, 11.91651749001402, 12.60174210332792, 13.21495713217107
