| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 6·11-s − 2·12-s + 16-s − 3·17-s − 18-s + 3·19-s − 6·22-s + 2·24-s − 25-s + 4·27-s − 32-s − 12·33-s + 3·34-s + 36-s − 3·38-s + 18·41-s − 11·43-s + 6·44-s − 2·48-s + 5·49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.688·19-s − 1.27·22-s + 0.408·24-s − 1/5·25-s + 0.769·27-s − 0.176·32-s − 2.08·33-s + 0.514·34-s + 1/6·36-s − 0.486·38-s + 2.81·41-s − 1.67·43-s + 0.904·44-s − 0.288·48-s + 5/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7030580034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7030580034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 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
−9.860679846322321186541383805503, −9.260395385441702342114612270569, −9.060189657378483814596055928063, −8.572339642776222471631853842305, −7.63607089515176500236878656165, −7.42102597304711708794077996365, −6.63235663851031478053804468469, −6.16919205179629571276318192615, −6.04250937186358541719539543439, −5.08915542710075922488398693595, −4.52043703416240218185855183933, −3.82116156184430759877255384782, −2.97255420057433764240313570604, −1.83132368292549670790382570622, −0.843549464977689947392351107918,
0.843549464977689947392351107918, 1.83132368292549670790382570622, 2.97255420057433764240313570604, 3.82116156184430759877255384782, 4.52043703416240218185855183933, 5.08915542710075922488398693595, 6.04250937186358541719539543439, 6.16919205179629571276318192615, 6.63235663851031478053804468469, 7.42102597304711708794077996365, 7.63607089515176500236878656165, 8.572339642776222471631853842305, 9.060189657378483814596055928063, 9.260395385441702342114612270569, 9.860679846322321186541383805503
