| L(s) = 1 | + 2·3-s + 4-s + 6·5-s − 3·9-s − 6·11-s + 2·12-s + 12·15-s + 16-s + 6·20-s − 6·23-s + 17·25-s − 14·27-s + 10·31-s − 12·33-s − 3·36-s − 20·37-s − 6·44-s − 18·45-s + 24·47-s + 2·48-s + 2·49-s + 18·53-s − 36·55-s + 24·59-s + 12·60-s + 64-s − 8·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 2.68·5-s − 9-s − 1.80·11-s + 0.577·12-s + 3.09·15-s + 1/4·16-s + 1.34·20-s − 1.25·23-s + 17/5·25-s − 2.69·27-s + 1.79·31-s − 2.08·33-s − 1/2·36-s − 3.28·37-s − 0.904·44-s − 2.68·45-s + 3.50·47-s + 0.288·48-s + 2/7·49-s + 2.47·53-s − 4.85·55-s + 3.12·59-s + 1.54·60-s + 1/8·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3833764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3833764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.325723881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.325723881\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−7.36410689745433254804873728837, −7.14291882547162255627816684909, −6.54762216250182029125237453136, −5.94744023022306454130283790935, −5.75671469286390333326325976946, −5.48197881161062206485371067956, −5.35465835358337466615572727135, −4.57510660708520700229807955350, −3.79572593468297191488661918640, −3.31534392384618004485979133214, −2.61647407804135209006919461904, −2.51994269176562419526281033451, −2.11801128984211370773581265243, −1.88938183756548545374913634243, −0.69686121807223499961036044669,
0.69686121807223499961036044669, 1.88938183756548545374913634243, 2.11801128984211370773581265243, 2.51994269176562419526281033451, 2.61647407804135209006919461904, 3.31534392384618004485979133214, 3.79572593468297191488661918640, 4.57510660708520700229807955350, 5.35465835358337466615572727135, 5.48197881161062206485371067956, 5.75671469286390333326325976946, 5.94744023022306454130283790935, 6.54762216250182029125237453136, 7.14291882547162255627816684909, 7.36410689745433254804873728837
