| L(s) = 1 | − 3-s − 2·4-s − 3·7-s − 2·9-s + 2·12-s − 4·13-s + 3·21-s − 25-s + 5·27-s + 6·28-s − 9·31-s + 4·36-s + 3·37-s + 4·39-s − 2·43-s + 2·49-s + 8·52-s + 14·61-s + 6·63-s + 8·64-s + 7·67-s − 5·73-s + 75-s − 13·79-s + 81-s − 6·84-s + 12·91-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.13·7-s − 2/3·9-s + 0.577·12-s − 1.10·13-s + 0.654·21-s − 1/5·25-s + 0.962·27-s + 1.13·28-s − 1.61·31-s + 2/3·36-s + 0.493·37-s + 0.640·39-s − 0.304·43-s + 2/7·49-s + 1.10·52-s + 1.79·61-s + 0.755·63-s + 64-s + 0.855·67-s − 0.585·73-s + 0.115·75-s − 1.46·79-s + 1/9·81-s − 0.654·84-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−11.44143512570500340082196557809, −10.91781731857307716314603794114, −10.12524466987510967656593114869, −9.695234107713111107390157126114, −9.248006964914772692335807990868, −8.660966236514361236858631650573, −7.994769985244196073328035137928, −7.08328881839961312359819628366, −6.64271894833807211348509290342, −5.68680949131679008995515836152, −5.30891567797866767032422615638, −4.44698021248356841723306301485, −3.60724543959152583930898114695, −2.58257775926797291772517437762, 0,
2.58257775926797291772517437762, 3.60724543959152583930898114695, 4.44698021248356841723306301485, 5.30891567797866767032422615638, 5.68680949131679008995515836152, 6.64271894833807211348509290342, 7.08328881839961312359819628366, 7.994769985244196073328035137928, 8.660966236514361236858631650573, 9.248006964914772692335807990868, 9.695234107713111107390157126114, 10.12524466987510967656593114869, 10.91781731857307716314603794114, 11.44143512570500340082196557809
