| L(s) = 1 | − 3·2-s + 3·3-s + 4·4-s + 6·5-s − 9·6-s − 3·8-s + 6·9-s − 18·10-s + 12·12-s + 3·13-s + 18·15-s + 3·16-s − 3·17-s − 18·18-s + 9·19-s + 24·20-s − 9·24-s + 17·25-s − 9·26-s + 9·27-s − 9·29-s − 54·30-s + 6·31-s − 6·32-s + 9·34-s + 24·36-s − 7·37-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 2·4-s + 2.68·5-s − 3.67·6-s − 1.06·8-s + 2·9-s − 5.69·10-s + 3.46·12-s + 0.832·13-s + 4.64·15-s + 3/4·16-s − 0.727·17-s − 4.24·18-s + 2.06·19-s + 5.36·20-s − 1.83·24-s + 17/5·25-s − 1.76·26-s + 1.73·27-s − 1.67·29-s − 9.85·30-s + 1.07·31-s − 1.06·32-s + 1.54·34-s + 4·36-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.783580157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783580157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T + 128 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71430142050717045548279579408, −10.70704290299500632395616395743, −9.966812068373971656602248906909, −9.622970219408949043899417321996, −9.489066931635769291659851217351, −9.339725434933373327355123457551, −8.758203295864613541670595723170, −8.508188353993272298851908614266, −7.87622207116831882661162770878, −7.55199664669973416981709926685, −6.95026303310152955015798327225, −6.34762994543872814409830464126, −5.88170551328094744574311991043, −5.33625661556151440208566839580, −4.60296041835022587449067178750, −3.38975232692628683184736471929, −3.10915928561486700508227992890, −2.18008246770824568750489151264, −1.68506314583233985598526436150, −1.31476918920905115884516405007,
1.31476918920905115884516405007, 1.68506314583233985598526436150, 2.18008246770824568750489151264, 3.10915928561486700508227992890, 3.38975232692628683184736471929, 4.60296041835022587449067178750, 5.33625661556151440208566839580, 5.88170551328094744574311991043, 6.34762994543872814409830464126, 6.95026303310152955015798327225, 7.55199664669973416981709926685, 7.87622207116831882661162770878, 8.508188353993272298851908614266, 8.758203295864613541670595723170, 9.339725434933373327355123457551, 9.489066931635769291659851217351, 9.622970219408949043899417321996, 9.966812068373971656602248906909, 10.70704290299500632395616395743, 10.71430142050717045548279579408
