L(s) = 1 | − 4·2-s + 8·4-s + 2·5-s − 8·8-s − 5·9-s − 8·10-s + 11-s + 8·13-s − 4·16-s + 20·18-s + 16·20-s − 4·22-s − 2·23-s − 7·25-s − 32·26-s + 32·32-s − 40·36-s − 16·40-s − 16·41-s + 8·44-s − 10·45-s + 8·46-s + 16·47-s − 10·49-s + 28·50-s + 64·52-s + 2·55-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 4·4-s + 0.894·5-s − 2.82·8-s − 5/3·9-s − 2.52·10-s + 0.301·11-s + 2.21·13-s − 16-s + 4.71·18-s + 3.57·20-s − 0.852·22-s − 0.417·23-s − 7/5·25-s − 6.27·26-s + 5.65·32-s − 6.66·36-s − 2.52·40-s − 2.49·41-s + 1.20·44-s − 1.49·45-s + 1.17·46-s + 2.33·47-s − 1.42·49-s + 3.95·50-s + 8.87·52-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480491 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480491 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$ | \( 1 - T \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.603539619290756001226038948684, −8.271456646397008703308309286986, −7.57431385562905041991919174068, −7.35497151208830631745730081452, −6.38571791919663056094366244020, −6.36261389471308870138602900888, −5.82022083514319120460449276497, −5.31637329804068265476360676633, −4.42771266189087965806263330587, −3.71023016084926681627402828974, −3.06255388161967307495815251846, −2.08501708743688275104095928757, −1.78686819759584429785481099010, −1.00636184286304673122517790617, 0,
1.00636184286304673122517790617, 1.78686819759584429785481099010, 2.08501708743688275104095928757, 3.06255388161967307495815251846, 3.71023016084926681627402828974, 4.42771266189087965806263330587, 5.31637329804068265476360676633, 5.82022083514319120460449276497, 6.36261389471308870138602900888, 6.38571791919663056094366244020, 7.35497151208830631745730081452, 7.57431385562905041991919174068, 8.271456646397008703308309286986, 8.603539619290756001226038948684