| L(s) = 1 | − 40·11-s + 8·17-s + 72·19-s + 28·25-s + 8·41-s + 8·43-s + 100·49-s + 88·59-s − 216·67-s − 88·73-s − 40·83-s + 216·89-s − 328·97-s − 424·107-s + 312·113-s + 660·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 540·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3.63·11-s + 8/17·17-s + 3.78·19-s + 1.11·25-s + 8/41·41-s + 8/43·43-s + 2.04·49-s + 1.49·59-s − 3.22·67-s − 1.20·73-s − 0.481·83-s + 2.42·89-s − 3.38·97-s − 3.96·107-s + 2.76·113-s + 5.45·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.19·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.241893253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.241893253\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 934 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5254 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 20 T + 270 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 540 T^{2} + 125414 T^{4} - 540 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 454 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 36 T + 1038 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 484 T^{2} + 517894 T^{4} - 484 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 988 T^{2} + 1510630 T^{4} - 988 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2140 T^{2} + 4266022 T^{4} - 2140 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 2854 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 1390 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7428 T^{2} + 23258246 T^{4} - 7428 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10204 T^{2} + 41749414 T^{4} - 10204 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 44 T + 6478 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 796 T^{2} - 6495386 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 108 T + 11822 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18276 T^{2} + 133865606 T^{4} - 18276 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 44 T + 9990 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1028 T^{2} + 28324230 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 20 T + 2926 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 108 T + 15558 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 164 T + 22342 T^{2} + 164 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−5.94056229119152633150304105712, −5.93052562539039821916362470584, −5.76601051506821331728513809780, −5.43198461365032634260418817809, −5.35598174836839907981909303041, −5.17701065519210138869162983895, −5.14700587991101769356064523713, −4.90514712338950880177739459052, −4.55837929306307406001396538686, −4.35090144753147821377807822985, −4.10022501872830995270130090367, −3.81024389720489197490987398561, −3.44744971523402868544222301472, −3.22740294330906945425934732254, −3.17415157186922409851518203271, −2.73297342910185675437403467902, −2.64330274121428607418480361820, −2.55996539578879918508988376091, −2.37730695286283862643170946684, −1.79261203019023040640220893156, −1.29074940513651716218377819777, −1.27459521408495924607353258048, −0.976898943014320843354289391385, −0.50410706476296196122522386337, −0.16247275539268949558832027521,
0.16247275539268949558832027521, 0.50410706476296196122522386337, 0.976898943014320843354289391385, 1.27459521408495924607353258048, 1.29074940513651716218377819777, 1.79261203019023040640220893156, 2.37730695286283862643170946684, 2.55996539578879918508988376091, 2.64330274121428607418480361820, 2.73297342910185675437403467902, 3.17415157186922409851518203271, 3.22740294330906945425934732254, 3.44744971523402868544222301472, 3.81024389720489197490987398561, 4.10022501872830995270130090367, 4.35090144753147821377807822985, 4.55837929306307406001396538686, 4.90514712338950880177739459052, 5.14700587991101769356064523713, 5.17701065519210138869162983895, 5.35598174836839907981909303041, 5.43198461365032634260418817809, 5.76601051506821331728513809780, 5.93052562539039821916362470584, 5.94056229119152633150304105712
