| L(s) = 1 | − 1.34e3·25-s − 9.60e3·49-s − 4.22e4·73-s + 7.48e4·97-s − 5.85e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 956·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 2.15·25-s − 4·49-s − 7.92·73-s + 7.95·97-s − 4·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 0.0334·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + 2.01e−5·223-s + 1.94e−5·227-s + 1.90e−5·229-s + 1.84e−5·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5235229571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5235229571\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 672 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 240 T + p^{4} T^{2} )^{2}( 1 + 240 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 154560 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 137760 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2162 T + p^{4} T^{2} )^{2}( 1 + 2162 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 4374720 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 12509280 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6958 T + p^{4} T^{2} )^{2}( 1 + 6958 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10560 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 121779840 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18720 T + p^{4} T^{2} )^{4} \) |
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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
−6.28495669890550565822269418133, −6.21015756475412544170426355625, −6.16565933640150418547731725825, −5.67786881080481962472761600965, −5.63167722324083023148001792703, −5.45409820866227802647304204622, −4.95752428243568856188912211214, −4.76754172026296237085763446604, −4.57872632594766373833306585339, −4.39324170780657747469194635085, −4.36162662501495689811360757652, −3.72090650427857422115803309410, −3.58777964574750556314368066556, −3.33205073283063834439939351665, −3.29969303403021287285733801061, −2.88371071159722253457762453337, −2.54052020932108556854317978005, −2.26954513349019227248428144420, −2.04660559822061848755071804033, −1.63312934763286789660563969823, −1.42182165187620461634407258179, −1.32852444151702308084572960289, −0.846624762118022294021181763905, −0.23716706982095321213546668752, −0.15238773335936143966685413887,
0.15238773335936143966685413887, 0.23716706982095321213546668752, 0.846624762118022294021181763905, 1.32852444151702308084572960289, 1.42182165187620461634407258179, 1.63312934763286789660563969823, 2.04660559822061848755071804033, 2.26954513349019227248428144420, 2.54052020932108556854317978005, 2.88371071159722253457762453337, 3.29969303403021287285733801061, 3.33205073283063834439939351665, 3.58777964574750556314368066556, 3.72090650427857422115803309410, 4.36162662501495689811360757652, 4.39324170780657747469194635085, 4.57872632594766373833306585339, 4.76754172026296237085763446604, 4.95752428243568856188912211214, 5.45409820866227802647304204622, 5.63167722324083023148001792703, 5.67786881080481962472761600965, 6.16565933640150418547731725825, 6.21015756475412544170426355625, 6.28495669890550565822269418133
