L(s) = 1 | + 3-s − 2·4-s − 3·5-s − 2·9-s + 2·11-s − 2·12-s − 3·15-s + 6·20-s − 12·23-s − 25-s − 2·27-s + 31-s + 2·33-s + 4·36-s − 14·37-s − 4·44-s + 6·45-s + 10·47-s − 5·49-s + 7·53-s − 6·55-s + 6·60-s + 8·64-s + 3·67-s − 12·69-s + 71-s − 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s + 0.603·11-s − 0.577·12-s − 0.774·15-s + 1.34·20-s − 2.50·23-s − 1/5·25-s − 0.384·27-s + 0.179·31-s + 0.348·33-s + 2/3·36-s − 2.30·37-s − 0.603·44-s + 0.894·45-s + 1.45·47-s − 5/7·49-s + 0.961·53-s − 0.809·55-s + 0.774·60-s + 64-s + 0.366·67-s − 1.44·69-s + 0.118·71-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21417 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21417 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−10.53689725368450372854599592141, −9.826822870483360437079190157580, −9.445421472012554194147888312809, −8.639862454686285533634005985397, −8.505212176467540948543859627828, −8.001364511684348274511215868827, −7.42540434027782802193190640285, −6.71321116389340966816178045003, −5.89217844221131912664123822930, −5.25273770817460594855381165623, −4.26184914397204860940102278366, −3.96514181338040128750591760867, −3.38791302940571504979288791714, −2.14227175602667477107494673109, 0,
2.14227175602667477107494673109, 3.38791302940571504979288791714, 3.96514181338040128750591760867, 4.26184914397204860940102278366, 5.25273770817460594855381165623, 5.89217844221131912664123822930, 6.71321116389340966816178045003, 7.42540434027782802193190640285, 8.001364511684348274511215868827, 8.505212176467540948543859627828, 8.639862454686285533634005985397, 9.445421472012554194147888312809, 9.826822870483360437079190157580, 10.53689725368450372854599592141