L(s) = 1 | − 3-s + 2·5-s − 2·7-s − 2·9-s − 2·15-s − 4·16-s − 4·17-s + 2·21-s − 7·25-s + 5·27-s − 4·35-s + 6·37-s − 16·41-s − 12·43-s − 4·45-s + 16·47-s + 4·48-s − 3·49-s + 4·51-s + 10·59-s + 4·63-s − 14·67-s + 7·75-s − 20·79-s − 8·80-s + 81-s − 12·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s − 2/3·9-s − 0.516·15-s − 16-s − 0.970·17-s + 0.436·21-s − 7/5·25-s + 0.962·27-s − 0.676·35-s + 0.986·37-s − 2.49·41-s − 1.82·43-s − 0.596·45-s + 2.33·47-s + 0.577·48-s − 3/7·49-s + 0.560·51-s + 1.30·59-s + 0.503·63-s − 1.71·67-s + 0.808·75-s − 2.25·79-s − 0.894·80-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 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
−10.03550909718107888433464868208, −9.306254765286992650868144024138, −8.701526732425662214712394784166, −8.603539619290756001226038948684, −7.59708149153402218158879958117, −7.03477908276603020210779304212, −6.36261389471308870138602900888, −6.21101030419725238522849385481, −5.55294080899685406164716506758, −4.97685157209324421449824776256, −4.28111751945710160622264923994, −3.43654811333543968800336820680, −2.58331042621701060186581520596, −1.87316747898306017870215940641, 0,
1.87316747898306017870215940641, 2.58331042621701060186581520596, 3.43654811333543968800336820680, 4.28111751945710160622264923994, 4.97685157209324421449824776256, 5.55294080899685406164716506758, 6.21101030419725238522849385481, 6.36261389471308870138602900888, 7.03477908276603020210779304212, 7.59708149153402218158879958117, 8.603539619290756001226038948684, 8.701526732425662214712394784166, 9.306254765286992650868144024138, 10.03550909718107888433464868208