L(s) = 1 | − 2·3-s − 3·4-s − 6·7-s + 9-s + 6·12-s + 5·13-s + 5·16-s − 9·19-s + 12·21-s + 25-s + 4·27-s + 18·28-s + 2·31-s − 3·36-s − 16·37-s − 10·39-s − 3·43-s − 10·48-s + 17·49-s − 15·52-s + 18·57-s − 6·61-s − 6·63-s − 3·64-s − 67-s + 5·73-s − 2·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 3/2·4-s − 2.26·7-s + 1/3·9-s + 1.73·12-s + 1.38·13-s + 5/4·16-s − 2.06·19-s + 2.61·21-s + 1/5·25-s + 0.769·27-s + 3.40·28-s + 0.359·31-s − 1/2·36-s − 2.63·37-s − 1.60·39-s − 0.457·43-s − 1.44·48-s + 17/7·49-s − 2.08·52-s + 2.38·57-s − 0.768·61-s − 0.755·63-s − 3/8·64-s − 0.122·67-s + 0.585·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 379 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 11 T + p T^{2} ) \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−12.52904316215616694165779948272, −12.05468628216561216405853219287, −11.02103899512350083464916962731, −10.47881770793489996875163931921, −10.09884140552019139191044688156, −9.298818815832352824793294835629, −8.761020043918534300736028943085, −8.368184913151009783371565215147, −6.84996672400503775780963825268, −6.41170868431526283729680636959, −5.89846952311957618693090452483, −5.01094266926562858529869899609, −4.03936375515308084295939588309, −3.34254052952856063975225663394, 0,
3.34254052952856063975225663394, 4.03936375515308084295939588309, 5.01094266926562858529869899609, 5.89846952311957618693090452483, 6.41170868431526283729680636959, 6.84996672400503775780963825268, 8.368184913151009783371565215147, 8.761020043918534300736028943085, 9.298818815832352824793294835629, 10.09884140552019139191044688156, 10.47881770793489996875163931921, 11.02103899512350083464916962731, 12.05468628216561216405853219287, 12.52904316215616694165779948272