| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s − 12·17-s − 6·23-s − 25-s + 4·27-s + 6·29-s − 3·36-s + 2·43-s + 2·48-s + 10·49-s − 24·51-s − 12·53-s + 4·61-s − 64-s + 12·68-s − 12·69-s − 2·75-s + 10·79-s + 5·81-s + 12·87-s + 6·92-s + 100-s − 12·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s − 2.91·17-s − 1.25·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s − 1/2·36-s + 0.304·43-s + 0.288·48-s + 10/7·49-s − 3.36·51-s − 1.64·53-s + 0.512·61-s − 1/8·64-s + 1.45·68-s − 1.44·69-s − 0.230·75-s + 1.12·79-s + 5/9·81-s + 1.28·87-s + 0.625·92-s + 1/10·100-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.759956498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759956498\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.349542012464486739397088149720, −8.326249008305314558683950913306, −7.79554238499848118647564076581, −7.39853799603499075590438023952, −6.92915391690841619223447171287, −6.76488647011071174747610385004, −6.25230391679810651195193028337, −6.05740187538049644759979417788, −5.44939507567546335458426373303, −4.95852894655401897808059470391, −4.47503395124674126951283021363, −4.42947053621484210932482280760, −3.84959798874664381953104334032, −3.72825288040742446441310192257, −2.86257019471197360452941215465, −2.73372979041411725114475866718, −2.04866679336458395753259929269, −1.97452111022205197322206413750, −1.15658001409719249031054745265, −0.33373083231994877022887768936,
0.33373083231994877022887768936, 1.15658001409719249031054745265, 1.97452111022205197322206413750, 2.04866679336458395753259929269, 2.73372979041411725114475866718, 2.86257019471197360452941215465, 3.72825288040742446441310192257, 3.84959798874664381953104334032, 4.42947053621484210932482280760, 4.47503395124674126951283021363, 4.95852894655401897808059470391, 5.44939507567546335458426373303, 6.05740187538049644759979417788, 6.25230391679810651195193028337, 6.76488647011071174747610385004, 6.92915391690841619223447171287, 7.39853799603499075590438023952, 7.79554238499848118647564076581, 8.326249008305314558683950913306, 8.349542012464486739397088149720
