| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 3·8-s + 3·9-s − 11-s + 2·12-s − 5·13-s + 16-s − 2·17-s + 3·18-s + 3·19-s − 22-s + 9·23-s + 6·24-s − 5·26-s + 4·27-s − 2·31-s − 32-s − 2·33-s − 2·34-s + 3·36-s + 2·37-s + 3·38-s − 10·39-s − 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.06·8-s + 9-s − 0.301·11-s + 0.577·12-s − 1.38·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.688·19-s − 0.213·22-s + 1.87·23-s + 1.22·24-s − 0.980·26-s + 0.769·27-s − 0.359·31-s − 0.176·32-s − 0.348·33-s − 0.342·34-s + 1/2·36-s + 0.328·37-s + 0.486·38-s − 1.60·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.774171518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.774171518\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.764253151559417429339830746772, −9.444323961896695720597510882329, −9.070768520309426003266446484239, −8.794408196397940696107011062064, −8.048399544365154624176207793136, −7.78490445916490596566433996797, −7.50321201559157969188142989282, −6.89064354706775396196392493064, −6.88675509768582542808761554616, −6.26467400542289023859256128555, −5.28625657918567498754876508693, −5.26959184189309864969510416770, −4.68358104130766192200494035282, −4.38735719017966490173412686704, −3.72588257118657684087442847846, −3.23978458671317793614731250644, −2.75288785604221436384025836454, −2.26583470835850821177697685092, −1.80325434092532697817968423072, −0.845061288028008947204565560409,
0.845061288028008947204565560409, 1.80325434092532697817968423072, 2.26583470835850821177697685092, 2.75288785604221436384025836454, 3.23978458671317793614731250644, 3.72588257118657684087442847846, 4.38735719017966490173412686704, 4.68358104130766192200494035282, 5.26959184189309864969510416770, 5.28625657918567498754876508693, 6.26467400542289023859256128555, 6.88675509768582542808761554616, 6.89064354706775396196392493064, 7.50321201559157969188142989282, 7.78490445916490596566433996797, 8.048399544365154624176207793136, 8.794408196397940696107011062064, 9.070768520309426003266446484239, 9.444323961896695720597510882329, 9.764253151559417429339830746772
