L(s) = 1 | − 3·3-s + 4-s − 6·7-s + 6·9-s − 9·11-s − 3·12-s + 16-s + 18·21-s + 2·25-s − 9·27-s − 6·28-s + 27·33-s + 6·36-s − 8·37-s + 14·41-s − 9·44-s − 6·47-s − 3·48-s + 14·49-s + 6·53-s − 36·63-s + 64-s + 23·67-s − 2·71-s + 13·73-s − 6·75-s + 54·77-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1/2·4-s − 2.26·7-s + 2·9-s − 2.71·11-s − 0.866·12-s + 1/4·16-s + 3.92·21-s + 2/5·25-s − 1.73·27-s − 1.13·28-s + 4.70·33-s + 36-s − 1.31·37-s + 2.18·41-s − 1.35·44-s − 0.875·47-s − 0.433·48-s + 2·49-s + 0.824·53-s − 4.53·63-s + 1/8·64-s + 2.80·67-s − 0.237·71-s + 1.52·73-s − 0.692·75-s + 6.15·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 153 T^{2} + p^{2} T^{4} \) |
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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
−8.085884637665208668443474394859, −7.86636121046339608929679800601, −7.05804968179849580755657181925, −7.04409636362076918775297120280, −6.30447611121309330314211247730, −6.18112871578502678243715278936, −5.55593180552572775639322259506, −5.16044247914585545884743049492, −4.87961619983670547867372044163, −3.87951592996168551662468150846, −3.39138665208870675445467987312, −2.68872003483472680365229309029, −2.25582489113105897164582804977, −0.73463450930227867455779333973, 0,
0.73463450930227867455779333973, 2.25582489113105897164582804977, 2.68872003483472680365229309029, 3.39138665208870675445467987312, 3.87951592996168551662468150846, 4.87961619983670547867372044163, 5.16044247914585545884743049492, 5.55593180552572775639322259506, 6.18112871578502678243715278936, 6.30447611121309330314211247730, 7.04409636362076918775297120280, 7.05804968179849580755657181925, 7.86636121046339608929679800601, 8.085884637665208668443474394859