| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s + 6·13-s − 14-s − 16-s − 2·17-s + 6·23-s − 5·25-s + 6·26-s + 28-s + 29-s − 4·31-s + 5·32-s − 2·34-s + 6·37-s − 10·41-s + 10·43-s + 6·46-s − 8·47-s + 49-s − 5·50-s − 6·52-s − 10·53-s + 3·56-s + 58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 1.25·23-s − 25-s + 1.17·26-s + 0.188·28-s + 0.185·29-s − 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.986·37-s − 1.56·41-s + 1.52·43-s + 0.884·46-s − 1.16·47-s + 1/7·49-s − 0.707·50-s − 0.832·52-s − 1.37·53-s + 0.400·56-s + 0.131·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.480510581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480510581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.20433447799719, −12.65994560168846, −12.15971594465469, −11.53951335652383, −11.21381354113945, −10.80214258418667, −10.11956989317996, −9.628072957499288, −9.177259556751238, −8.787243530859665, −8.284312369644692, −7.904082660564723, −7.030077107090220, −6.673378695809450, −6.076460603709806, −5.772981058646297, −5.188481459648613, −4.666175417197322, −4.064377825530238, −3.667225328445684, −3.231004011424254, −2.638600035854642, −1.839844697251585, −1.103484637916807, −0.4322211726299848,
0.4322211726299848, 1.103484637916807, 1.839844697251585, 2.638600035854642, 3.231004011424254, 3.667225328445684, 4.064377825530238, 4.666175417197322, 5.188481459648613, 5.772981058646297, 6.076460603709806, 6.673378695809450, 7.030077107090220, 7.904082660564723, 8.284312369644692, 8.787243530859665, 9.177259556751238, 9.628072957499288, 10.11956989317996, 10.80214258418667, 11.21381354113945, 11.53951335652383, 12.15971594465469, 12.65994560168846, 13.20433447799719
