| L(s) = 1 | + 208·7-s − 732·13-s − 256·16-s + 2.71e4·31-s − 1.48e4·37-s − 1.27e4·43-s + 2.16e4·49-s + 5.49e4·61-s − 1.36e5·67-s + 1.19e5·73-s − 1.52e5·91-s − 3.84e5·97-s + 8.21e4·103-s − 5.32e4·112-s + 6.13e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.67e5·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.60·7-s − 1.20·13-s − 1/4·16-s + 5.07·31-s − 1.78·37-s − 1.04·43-s + 1.28·49-s + 1.89·61-s − 3.70·67-s + 2.62·73-s − 1.92·91-s − 4.14·97-s + 0.762·103-s − 0.401·112-s + 3.80·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.721·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.636128815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.636128815\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 104 T + 5408 T^{2} - 104 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2534 p^{2} T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 366 T + 66978 T^{2} + 366 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 2695281514306 T^{4} + p^{20} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 508534 T^{2} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 43180786439198 T^{4} + p^{20} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 7962904 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6784 T + p^{5} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 7446 T + 27721458 T^{2} + 7446 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 82655024 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6360 T + 20224800 T^{2} + 6360 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 9133371392128898 T^{4} + p^{20} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 306323542400025998 T^{4} + p^{20} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 144806390 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13740 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 68128 T + 2320712192 T^{2} + 68128 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1301352110 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 59782 T + 1786943762 T^{2} - 59782 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 2634301214 T^{2} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 6332652458657888494 T^{4} + p^{20} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 10877569280 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 192186 T + 18467729298 T^{2} + 192186 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−7.06567964675805913195739350332, −6.92780746746238174109714844667, −6.78046056911968728777764182727, −6.60749856914098908017679712495, −6.16522217850273999102494474260, −5.87198585669582634560332588210, −5.62502955408275315206639871243, −5.47181328543383787671046626957, −4.95237443059099131570385810800, −4.84865497172954611395228692688, −4.67515670771307342481076212185, −4.43011102914484687437254467215, −4.29773005788543456470195232346, −3.90514324453486619891727921943, −3.38811373625323384211773989357, −3.08465899103667496539752394640, −2.83279513606061245539069270527, −2.71508134970827262013081669291, −2.07054086806778998418539445143, −1.95634958149178792469295721614, −1.76433654560856909683308097588, −1.16778263234508688776017036118, −0.980505314877866470825167621684, −0.48979524782549464151845284154, −0.41831385284723725214229965921,
0.41831385284723725214229965921, 0.48979524782549464151845284154, 0.980505314877866470825167621684, 1.16778263234508688776017036118, 1.76433654560856909683308097588, 1.95634958149178792469295721614, 2.07054086806778998418539445143, 2.71508134970827262013081669291, 2.83279513606061245539069270527, 3.08465899103667496539752394640, 3.38811373625323384211773989357, 3.90514324453486619891727921943, 4.29773005788543456470195232346, 4.43011102914484687437254467215, 4.67515670771307342481076212185, 4.84865497172954611395228692688, 4.95237443059099131570385810800, 5.47181328543383787671046626957, 5.62502955408275315206639871243, 5.87198585669582634560332588210, 6.16522217850273999102494474260, 6.60749856914098908017679712495, 6.78046056911968728777764182727, 6.92780746746238174109714844667, 7.06567964675805913195739350332
