| L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 14·13-s − 9·19-s + 3·21-s − 5·25-s + 9·27-s − 15·31-s + 37-s + 42·39-s − 6·49-s − 27·57-s − 14·61-s + 6·63-s + 21·67-s − 7·73-s − 15·75-s − 21·79-s + 9·81-s + 14·91-s − 45·93-s − 28·97-s − 33·103-s + 17·109-s + 3·111-s + 84·117-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 3.88·13-s − 2.06·19-s + 0.654·21-s − 25-s + 1.73·27-s − 2.69·31-s + 0.164·37-s + 6.72·39-s − 6/7·49-s − 3.57·57-s − 1.79·61-s + 0.755·63-s + 2.56·67-s − 0.819·73-s − 1.73·75-s − 2.36·79-s + 81-s + 1.46·91-s − 4.66·93-s − 2.84·97-s − 3.25·103-s + 1.62·109-s + 0.284·111-s + 7.76·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.323838826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.323838826\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−11.56506801098004639560069405545, −11.16840852359451765119724649848, −10.73616364278287304953217711698, −10.67056682563100102171335089346, −9.733221932906258709951472011255, −9.292393667647362677410409950250, −8.710651413955247205602958665799, −8.666553837225620866397138182937, −8.015036189422552053944168052198, −8.005336318807721716619503830718, −6.95603385077621266170733743200, −6.64013876247236327421836003458, −5.86227972333237008020225043658, −5.63843513282617043377495200063, −4.21850698959596115677830892134, −4.10137997542137648858403093241, −3.59193014560405072238882888126, −2.98527943096114065119028565448, −1.72612189875925720973248939558, −1.67919957356378066769372483412,
1.67919957356378066769372483412, 1.72612189875925720973248939558, 2.98527943096114065119028565448, 3.59193014560405072238882888126, 4.10137997542137648858403093241, 4.21850698959596115677830892134, 5.63843513282617043377495200063, 5.86227972333237008020225043658, 6.64013876247236327421836003458, 6.95603385077621266170733743200, 8.005336318807721716619503830718, 8.015036189422552053944168052198, 8.666553837225620866397138182937, 8.710651413955247205602958665799, 9.292393667647362677410409950250, 9.733221932906258709951472011255, 10.67056682563100102171335089346, 10.73616364278287304953217711698, 11.16840852359451765119724649848, 11.56506801098004639560069405545
