| L(s) = 1 | + 4-s + 2·7-s + 4·9-s − 3·16-s − 7·19-s − 3·23-s − 8·25-s + 2·28-s + 2·29-s + 4·36-s − 9·37-s + 41-s + 12·43-s + 11·47-s − 10·49-s − 23·53-s − 59-s − 6·61-s + 8·63-s − 7·64-s − 7·76-s + 7·81-s − 2·83-s − 3·92-s + 2·97-s − 8·100-s − 5·101-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 4/3·9-s − 3/4·16-s − 1.60·19-s − 0.625·23-s − 8/5·25-s + 0.377·28-s + 0.371·29-s + 2/3·36-s − 1.47·37-s + 0.156·41-s + 1.82·43-s + 1.60·47-s − 1.42·49-s − 3.15·53-s − 0.130·59-s − 0.768·61-s + 1.00·63-s − 7/8·64-s − 0.802·76-s + 7/9·81-s − 0.219·83-s − 0.312·92-s + 0.203·97-s − 4/5·100-s − 0.497·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 892607 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 892607 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 197 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.979742291153288781871848893182, −7.53122752481409051572118846440, −7.11649517094351816423407014922, −6.64977157586489266233560120173, −6.19348038832697690675671870268, −5.91988019515216557786402513082, −5.13586660586124956397330252434, −4.63762945663876974479660843080, −4.22338616425628103051953391187, −3.98034143267992708286463409759, −3.13703035520467844146450726442, −2.34264573328443547401397917597, −1.85859105117832085629512939522, −1.47114312704971528478173872682, 0,
1.47114312704971528478173872682, 1.85859105117832085629512939522, 2.34264573328443547401397917597, 3.13703035520467844146450726442, 3.98034143267992708286463409759, 4.22338616425628103051953391187, 4.63762945663876974479660843080, 5.13586660586124956397330252434, 5.91988019515216557786402513082, 6.19348038832697690675671870268, 6.64977157586489266233560120173, 7.11649517094351816423407014922, 7.53122752481409051572118846440, 7.979742291153288781871848893182
