L(s) = 1 | + 2-s + 2·3-s + 2·4-s + 2·6-s − 2·7-s + 5·8-s + 3·9-s − 2·11-s + 4·12-s − 2·13-s − 2·14-s + 5·16-s − 6·17-s + 3·18-s − 4·19-s − 4·21-s − 2·22-s + 2·23-s + 10·24-s − 2·26-s + 4·27-s − 4·28-s + 2·29-s − 2·31-s + 10·32-s − 4·33-s − 6·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 4-s + 0.816·6-s − 0.755·7-s + 1.76·8-s + 9-s − 0.603·11-s + 1.15·12-s − 0.554·13-s − 0.534·14-s + 5/4·16-s − 1.45·17-s + 0.707·18-s − 0.917·19-s − 0.872·21-s − 0.426·22-s + 0.417·23-s + 2.04·24-s − 0.392·26-s + 0.769·27-s − 0.755·28-s + 0.371·29-s − 0.359·31-s + 1.76·32-s − 0.696·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33350625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33350625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.283658334\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.283658334\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 110 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 129 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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
−8.474488482482199247999781927299, −7.70935676772730314395657631840, −7.52451721303055805792350413210, −7.28248503101945853814761201446, −6.89888355717479577768646063098, −6.46734210383121629068220646273, −6.42760518470388660025804293608, −5.85118580359455104610255846593, −5.11963274563894511131504750552, −5.11266616425640761138173681536, −4.42901734059988168403680239493, −4.40340461306776061556222345749, −3.71190676461332059675438448282, −3.57215432253729357472842397096, −2.95425203736711203862040693690, −2.43378164829561393571288384002, −2.36800610064009426129857228553, −1.92154049140804054072943268368, −1.34711177873983303269355542394, −0.48105070852092504498161870187,
0.48105070852092504498161870187, 1.34711177873983303269355542394, 1.92154049140804054072943268368, 2.36800610064009426129857228553, 2.43378164829561393571288384002, 2.95425203736711203862040693690, 3.57215432253729357472842397096, 3.71190676461332059675438448282, 4.40340461306776061556222345749, 4.42901734059988168403680239493, 5.11266616425640761138173681536, 5.11963274563894511131504750552, 5.85118580359455104610255846593, 6.42760518470388660025804293608, 6.46734210383121629068220646273, 6.89888355717479577768646063098, 7.28248503101945853814761201446, 7.52451721303055805792350413210, 7.70935676772730314395657631840, 8.474488482482199247999781927299