| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s − 9-s − 4·14-s − 16-s − 18-s + 8·23-s − 25-s + 4·28-s − 16·29-s + 5·32-s + 36-s + 8·43-s + 8·46-s + 9·49-s − 50-s + 12·56-s − 16·58-s + 4·63-s + 7·64-s − 24·67-s − 16·71-s + 3·72-s − 16·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s − 1/3·9-s − 1.06·14-s − 1/4·16-s − 0.235·18-s + 1.66·23-s − 1/5·25-s + 0.755·28-s − 2.97·29-s + 0.883·32-s + 1/6·36-s + 1.21·43-s + 1.17·46-s + 9/7·49-s − 0.141·50-s + 1.60·56-s − 2.10·58-s + 0.503·63-s + 7/8·64-s − 2.93·67-s − 1.89·71-s + 0.353·72-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8956551507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8956551507\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.529191545559901772161672908918, −7.58690878068296409615361770297, −7.44650613070360544674900996200, −6.98343250261335189624663441048, −6.30016154540729114804741718930, −5.96482399441904194134246858349, −5.60012765914228129428939123847, −5.23217102393739526477107835497, −4.39796138274330613047167381357, −4.16055906737778175525619503418, −3.43437762522046048230109707694, −3.11112264835231661863275680309, −2.68722425491877388547939879713, −1.64975686792702448264439652141, −0.41066989354847086012362744940,
0.41066989354847086012362744940, 1.64975686792702448264439652141, 2.68722425491877388547939879713, 3.11112264835231661863275680309, 3.43437762522046048230109707694, 4.16055906737778175525619503418, 4.39796138274330613047167381357, 5.23217102393739526477107835497, 5.60012765914228129428939123847, 5.96482399441904194134246858349, 6.30016154540729114804741718930, 6.98343250261335189624663441048, 7.44650613070360544674900996200, 7.58690878068296409615361770297, 8.529191545559901772161672908918
