L(s) = 1 | − 2·3-s − 4-s + 7-s + 9-s + 2·12-s − 13-s − 3·16-s + 19-s − 2·21-s + 25-s + 4·27-s − 28-s − 4·31-s − 36-s − 37-s + 2·39-s + 21·43-s + 6·48-s − 6·49-s + 52-s − 2·57-s − 6·61-s + 63-s + 7·64-s − 14·67-s − 24·73-s − 2·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 0.377·7-s + 1/3·9-s + 0.577·12-s − 0.277·13-s − 3/4·16-s + 0.229·19-s − 0.436·21-s + 1/5·25-s + 0.769·27-s − 0.188·28-s − 0.718·31-s − 1/6·36-s − 0.164·37-s + 0.320·39-s + 3.20·43-s + 0.866·48-s − 6/7·49-s + 0.138·52-s − 0.264·57-s − 0.768·61-s + 0.125·63-s + 7/8·64-s − 1.71·67-s − 2.80·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 1021 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 5 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
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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
−8.527112768906894629933695767071, −7.71027539671950869578442196374, −7.32768090390043634698694615176, −7.15160828459659687247487046166, −6.25481759128758121221925603570, −5.96180700820689622984031185083, −5.66159047286633629585668767024, −4.91042228597480187036882189665, −4.60266738586593880697487893071, −4.28515892487817675394568644103, −3.41307673535432304830602697543, −2.75581901588542386975307401680, −1.96969768125309672920786382233, −1.00932916595600540206105410876, 0,
1.00932916595600540206105410876, 1.96969768125309672920786382233, 2.75581901588542386975307401680, 3.41307673535432304830602697543, 4.28515892487817675394568644103, 4.60266738586593880697487893071, 4.91042228597480187036882189665, 5.66159047286633629585668767024, 5.96180700820689622984031185083, 6.25481759128758121221925603570, 7.15160828459659687247487046166, 7.32768090390043634698694615176, 7.71027539671950869578442196374, 8.527112768906894629933695767071