| L(s) = 1 | + 2-s + 4-s + 8-s + 2·9-s + 8·13-s + 16-s + 2·17-s + 2·18-s − 12·19-s − 6·25-s + 8·26-s + 32-s + 2·34-s + 2·36-s − 12·38-s + 12·43-s − 16·47-s + 2·49-s − 6·50-s + 8·52-s + 8·53-s − 4·59-s + 64-s + 4·67-s + 2·68-s + 2·72-s − 12·76-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s + 2.21·13-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 2.75·19-s − 6/5·25-s + 1.56·26-s + 0.176·32-s + 0.342·34-s + 1/3·36-s − 1.94·38-s + 1.82·43-s − 2.33·47-s + 2/7·49-s − 0.848·50-s + 1.10·52-s + 1.09·53-s − 0.520·59-s + 1/8·64-s + 0.488·67-s + 0.242·68-s + 0.235·72-s − 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.140055600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.140055600\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.60389981157573407446217034242, −9.966194783857914863126754522383, −9.266880437997485843191118396195, −8.712922597822970803022394832787, −8.149170853378179717399068124103, −7.83374365040642826619777615116, −6.88358390210605052670411460647, −6.39302249923793739922396430187, −6.08147959600045434700744078524, −5.43119973371138092443187760147, −4.46426130294909810305482256167, −3.99453713781203420250726150743, −3.58173899152872854729471411046, −2.37842004975190953552844472274, −1.49361093969928738645483145668,
1.49361093969928738645483145668, 2.37842004975190953552844472274, 3.58173899152872854729471411046, 3.99453713781203420250726150743, 4.46426130294909810305482256167, 5.43119973371138092443187760147, 6.08147959600045434700744078524, 6.39302249923793739922396430187, 6.88358390210605052670411460647, 7.83374365040642826619777615116, 8.149170853378179717399068124103, 8.712922597822970803022394832787, 9.266880437997485843191118396195, 9.966194783857914863126754522383, 10.60389981157573407446217034242
