L(s) = 1 | + 2-s − 4-s − 5-s − 8-s − 3·9-s − 10-s − 11-s + 6·13-s + 3·16-s − 3·18-s − 6·19-s + 20-s − 22-s − 8·23-s − 7·25-s + 6·26-s − 6·29-s + 3·32-s + 3·36-s − 6·38-s + 40-s + 44-s + 3·45-s − 8·46-s + 14·47-s − 5·49-s − 7·50-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.353·8-s − 9-s − 0.316·10-s − 0.301·11-s + 1.66·13-s + 3/4·16-s − 0.707·18-s − 1.37·19-s + 0.223·20-s − 0.213·22-s − 1.66·23-s − 7/5·25-s + 1.17·26-s − 1.11·29-s + 0.530·32-s + 1/2·36-s − 0.973·38-s + 0.158·40-s + 0.150·44-s + 0.447·45-s − 1.17·46-s + 2.04·47-s − 5/7·49-s − 0.989·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.254706501854957364501766642291, −8.907919440505075761265878776745, −8.309340271395688103631663104998, −8.042888692826423933797120299928, −7.62081938828138297229725147470, −6.69674548376858752943068105991, −6.03163169286647759642201268611, −5.77380920252080213203721246978, −5.37229546840605511929667692095, −4.29062859807934539822577422920, −4.03219376486958036593598966702, −3.64516090541421099591304301480, −2.70540793641000470403168683387, −1.72119349768487735074454778353, 0,
1.72119349768487735074454778353, 2.70540793641000470403168683387, 3.64516090541421099591304301480, 4.03219376486958036593598966702, 4.29062859807934539822577422920, 5.37229546840605511929667692095, 5.77380920252080213203721246978, 6.03163169286647759642201268611, 6.69674548376858752943068105991, 7.62081938828138297229725147470, 8.042888692826423933797120299928, 8.309340271395688103631663104998, 8.907919440505075761265878776745, 9.254706501854957364501766642291