| L(s) = 1 | + 2·7-s − 9-s + 8·17-s + 12·23-s + 10·25-s − 16·41-s − 8·47-s + 3·49-s − 2·63-s + 4·71-s + 4·73-s + 8·79-s + 81-s + 12·97-s + 16·103-s + 36·113-s + 16·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 24·161-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1/3·9-s + 1.94·17-s + 2.50·23-s + 2·25-s − 2.49·41-s − 1.16·47-s + 3/7·49-s − 0.251·63-s + 0.474·71-s + 0.468·73-s + 0.900·79-s + 1/9·81-s + 1.21·97-s + 1.57·103-s + 3.38·113-s + 1.46·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + 1.89·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.667227392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.667227392\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.482407267518918177009728247817, −8.110270550611922127619278200764, −7.55090942509323798610577960848, −7.28849655026615060729648131889, −6.89312458875876369594453513536, −6.83018988539595611799367683839, −6.00147689532617702060134140399, −5.98187034774515033539097406451, −5.28305722586905312944351402627, −5.02118610371483184584900692641, −4.72563675381539382616282028767, −4.71513313128057325711907366735, −3.66067661906793563971256821621, −3.33530731953269324470712018333, −3.20410245219993126917329645757, −2.77533863553591508732474637049, −1.99630062897520423637230473211, −1.63787614739273291174689575915, −0.894927630338655396370317804238, −0.76256401842454415998749725202,
0.76256401842454415998749725202, 0.894927630338655396370317804238, 1.63787614739273291174689575915, 1.99630062897520423637230473211, 2.77533863553591508732474637049, 3.20410245219993126917329645757, 3.33530731953269324470712018333, 3.66067661906793563971256821621, 4.71513313128057325711907366735, 4.72563675381539382616282028767, 5.02118610371483184584900692641, 5.28305722586905312944351402627, 5.98187034774515033539097406451, 6.00147689532617702060134140399, 6.83018988539595611799367683839, 6.89312458875876369594453513536, 7.28849655026615060729648131889, 7.55090942509323798610577960848, 8.110270550611922127619278200764, 8.482407267518918177009728247817
