| L(s) = 1 | + 2·3-s + 9-s + 2·13-s − 4·16-s + 14·17-s − 25-s − 4·27-s − 4·29-s + 4·39-s − 10·43-s − 8·48-s + 49-s + 28·51-s + 16·53-s + 8·61-s − 2·75-s + 14·79-s − 11·81-s − 8·87-s + 2·101-s − 24·103-s + 12·107-s − 8·113-s + 2·117-s − 2·121-s + 127-s − 20·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.554·13-s − 16-s + 3.39·17-s − 1/5·25-s − 0.769·27-s − 0.742·29-s + 0.640·39-s − 1.52·43-s − 1.15·48-s + 1/7·49-s + 3.92·51-s + 2.19·53-s + 1.02·61-s − 0.230·75-s + 1.57·79-s − 1.22·81-s − 0.857·87-s + 0.199·101-s − 2.36·103-s + 1.16·107-s − 0.752·113-s + 0.184·117-s − 0.181·121-s + 0.0887·127-s − 1.76·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.247421504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.247421504\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
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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
−9.662553758628742923229757766439, −9.349800992029380856054041667135, −8.694069126243437825975694888547, −8.334345867786345161330730273660, −7.77474048251910877902271437824, −7.51370223738528692248808465468, −6.83657059419669489748788694050, −6.14828682419663722224158809179, −5.36081557005507960233310195695, −5.27224721131239647063938857010, −3.97787308182029880377496144443, −3.66023265188208973537832346816, −3.05678979431387656308892016838, −2.27604132028694741351524060428, −1.26715152674933141322114489520,
1.26715152674933141322114489520, 2.27604132028694741351524060428, 3.05678979431387656308892016838, 3.66023265188208973537832346816, 3.97787308182029880377496144443, 5.27224721131239647063938857010, 5.36081557005507960233310195695, 6.14828682419663722224158809179, 6.83657059419669489748788694050, 7.51370223738528692248808465468, 7.77474048251910877902271437824, 8.334345867786345161330730273660, 8.694069126243437825975694888547, 9.349800992029380856054041667135, 9.662553758628742923229757766439
