| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 7-s − 4·8-s + 6·11-s + 2·12-s + 6·13-s − 2·14-s + 8·16-s − 4·17-s − 19-s + 21-s − 12·22-s − 4·23-s − 4·24-s − 12·26-s − 27-s + 2·28-s − 16·29-s − 31-s − 8·32-s + 6·33-s + 8·34-s + 7·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s − 1.41·8-s + 1.80·11-s + 0.577·12-s + 1.66·13-s − 0.534·14-s + 2·16-s − 0.970·17-s − 0.229·19-s + 0.218·21-s − 2.55·22-s − 0.834·23-s − 0.816·24-s − 2.35·26-s − 0.192·27-s + 0.377·28-s − 2.97·29-s − 0.179·31-s − 1.41·32-s + 1.04·33-s + 1.37·34-s + 1.15·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9957750609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9957750609\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−11.33015871726882319537788612982, −10.60326828591533869143859684439, −9.896477586106971616222271580205, −9.570936664165491272091133240933, −9.296518728297135060323926999916, −8.779706348119115643861110346472, −8.566069679677693509876507744596, −8.244995231051189764923982848980, −7.75420697950279746191706828978, −6.95826339055245779963999401317, −6.75133300900426866869539919183, −5.96571637295127254453832565960, −5.96171572878653631540120498057, −5.05407411604641121321106838386, −3.97996446654098130091831115625, −3.77254960248745380862309312713, −3.33679664056407299823865250386, −2.08269382618476784806709595124, −1.79706694307317425608173829921, −0.74154130201351039462854893281,
0.74154130201351039462854893281, 1.79706694307317425608173829921, 2.08269382618476784806709595124, 3.33679664056407299823865250386, 3.77254960248745380862309312713, 3.97996446654098130091831115625, 5.05407411604641121321106838386, 5.96171572878653631540120498057, 5.96571637295127254453832565960, 6.75133300900426866869539919183, 6.95826339055245779963999401317, 7.75420697950279746191706828978, 8.244995231051189764923982848980, 8.566069679677693509876507744596, 8.779706348119115643861110346472, 9.296518728297135060323926999916, 9.570936664165491272091133240933, 9.896477586106971616222271580205, 10.60326828591533869143859684439, 11.33015871726882319537788612982
