| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s − 4·11-s + 16-s − 7·17-s + 18-s − 5·19-s − 4·22-s + 2·25-s + 32-s − 7·34-s + 36-s − 5·38-s + 41-s + 43-s − 4·44-s − 10·49-s + 2·50-s + 14·59-s + 64-s + 67-s − 7·68-s + 72-s − 14·73-s − 5·76-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 1.14·19-s − 0.852·22-s + 2/5·25-s + 0.176·32-s − 1.20·34-s + 1/6·36-s − 0.811·38-s + 0.156·41-s + 0.152·43-s − 0.603·44-s − 1.42·49-s + 0.282·50-s + 1.82·59-s + 1/8·64-s + 0.122·67-s − 0.848·68-s + 0.117·72-s − 1.63·73-s − 0.573·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 125 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 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
−8.676150276878748543021743792401, −7.993946747440825662729138010891, −7.59078233103243078006434397694, −6.99691271872230242264973392872, −6.57508996859384239120140453053, −6.28506450348778634647487133835, −5.52702569556145910735051265614, −5.13347401823370244909730196223, −4.55981243678138553635450770017, −4.19803475597667648713979050261, −3.58944706565486261462988371382, −2.63075339273425998897552386087, −2.45376718092176215253822999058, −1.55267561460464788961978071183, 0,
1.55267561460464788961978071183, 2.45376718092176215253822999058, 2.63075339273425998897552386087, 3.58944706565486261462988371382, 4.19803475597667648713979050261, 4.55981243678138553635450770017, 5.13347401823370244909730196223, 5.52702569556145910735051265614, 6.28506450348778634647487133835, 6.57508996859384239120140453053, 6.99691271872230242264973392872, 7.59078233103243078006434397694, 7.993946747440825662729138010891, 8.676150276878748543021743792401
