| L(s) = 1 | − 2-s − 4-s − 7-s + 8-s + 4·9-s + 6·11-s + 14-s + 3·16-s − 4·18-s − 6·22-s − 9·23-s − 25-s + 28-s − 3·32-s − 4·36-s − 14·37-s + 9·43-s − 6·44-s + 9·46-s − 6·49-s + 50-s − 12·53-s − 56-s − 4·63-s − 5·64-s + 10·67-s + 21·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 0.353·8-s + 4/3·9-s + 1.80·11-s + 0.267·14-s + 3/4·16-s − 0.942·18-s − 1.27·22-s − 1.87·23-s − 1/5·25-s + 0.188·28-s − 0.530·32-s − 2/3·36-s − 2.30·37-s + 1.37·43-s − 0.904·44-s + 1.32·46-s − 6/7·49-s + 0.141·50-s − 1.64·53-s − 0.133·56-s − 0.503·63-s − 5/8·64-s + 1.22·67-s + 2.49·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5412741631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5412741631\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 8 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
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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
−12.40992743119000219830426930703, −12.02493042028340985141242906119, −11.19494436916396067256088003266, −10.41052341605502377252150880437, −9.844829833303857857340526608476, −9.505931410169624923239388157236, −8.994621533683078515979595306289, −8.227068586557118849738869505629, −7.62445480258900598893734797038, −6.71059934653250726444651732388, −6.31644324564184210493440834356, −5.19765492894545924172017700826, −4.07724382311546908680728359391, −3.71331134171430874660508363364, −1.61452194687764831160312520057,
1.61452194687764831160312520057, 3.71331134171430874660508363364, 4.07724382311546908680728359391, 5.19765492894545924172017700826, 6.31644324564184210493440834356, 6.71059934653250726444651732388, 7.62445480258900598893734797038, 8.227068586557118849738869505629, 8.994621533683078515979595306289, 9.505931410169624923239388157236, 9.844829833303857857340526608476, 10.41052341605502377252150880437, 11.19494436916396067256088003266, 12.02493042028340985141242906119, 12.40992743119000219830426930703
