| L(s) = 1 | − 18·9-s + 16·16-s − 74·19-s − 92·31-s − 52·37-s + 288·43-s − 23·49-s + 26·67-s + 194·73-s + 243·81-s + 338·97-s + 388·103-s + 286·109-s + 127-s + 131-s + 137-s + 139-s − 288·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + 1.33e3·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2·9-s + 16-s − 3.89·19-s − 2.96·31-s − 1.40·37-s + 6.69·43-s − 0.469·49-s + 0.388·67-s + 2.65·73-s + 3·81-s + 3.48·97-s + 3.76·103-s + 2.62·109-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s + 7.78·171-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.295051673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295051673\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 23 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 146 T^{2} + p^{4} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{2}( 1 + 647 T^{2} + p^{4} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2}( 1 + 1559 T^{2} + p^{4} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2}( 1 + 2591 T^{2} + p^{4} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )^{2}( 1 - 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 7199 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2}( 1 - 8809 T^{2} + p^{4} T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 97 T + p^{2} T^{2} )^{2}( 1 + 9791 T^{2} + p^{4} T^{4} ) \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 4679 T^{2} + p^{4} T^{4} )( 1 + 7682 T^{2} + p^{4} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 169 T + p^{2} T^{2} )^{2}( 1 - 18814 T^{2} + p^{4} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.590758541139868620295391236597, −8.093988524218395430170956815315, −8.082414675709755986941713213733, −7.56206213963845820000127761003, −7.44969458387412084664979733566, −7.31271768781510054562833018129, −6.88550193787768701126586177261, −6.35262339612550828513491399091, −6.15421681951296309905485742810, −6.05826739248020409634868229290, −5.86321404568592789714052415958, −5.67712692164981874051161344381, −5.13123311286222380908763297055, −4.79508433501567307136306910180, −4.76051312819637682341185702830, −4.08330024187137647324822711500, −3.77526501619268962002413109655, −3.63597200008768203033815993529, −3.45473962938802689151429214612, −2.55889965211316899987296848087, −2.42060520801050371605608707649, −2.22864723252144908688851328285, −1.81242638848222915123695785705, −0.76891382835306899847042272858, −0.36751499509788284979556185683,
0.36751499509788284979556185683, 0.76891382835306899847042272858, 1.81242638848222915123695785705, 2.22864723252144908688851328285, 2.42060520801050371605608707649, 2.55889965211316899987296848087, 3.45473962938802689151429214612, 3.63597200008768203033815993529, 3.77526501619268962002413109655, 4.08330024187137647324822711500, 4.76051312819637682341185702830, 4.79508433501567307136306910180, 5.13123311286222380908763297055, 5.67712692164981874051161344381, 5.86321404568592789714052415958, 6.05826739248020409634868229290, 6.15421681951296309905485742810, 6.35262339612550828513491399091, 6.88550193787768701126586177261, 7.31271768781510054562833018129, 7.44969458387412084664979733566, 7.56206213963845820000127761003, 8.082414675709755986941713213733, 8.093988524218395430170956815315, 8.590758541139868620295391236597
