| L(s) = 1 | + 32·7-s + 72·17-s − 512·23-s + 52·25-s + 160·31-s − 872·41-s − 448·47-s − 316·49-s − 4.09e3·71-s − 1.32e3·73-s − 992·79-s + 1.06e3·89-s − 2.44e3·97-s + 3.42e3·103-s − 456·113-s + 2.30e3·119-s − 1.36e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 1.63e4·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.72·7-s + 1.02·17-s − 4.64·23-s + 0.415·25-s + 0.926·31-s − 3.32·41-s − 1.39·47-s − 0.921·49-s − 6.84·71-s − 2.11·73-s − 1.41·79-s + 1.26·89-s − 2.55·97-s + 3.27·103-s − 0.379·113-s + 1.77·119-s − 1.02·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s − 8.02·161-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1219914104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1219914104\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 18614 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 16 T + 542 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 124 p T^{2} + 3795254 T^{4} + 124 p^{7} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4532 T^{2} + 10475286 T^{4} - 4532 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 36 T + 6822 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 11532 T^{2} + 65783830 T^{4} - 11532 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 256 T + 39886 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 52308 T^{2} + 1484422166 T^{4} - 52308 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 80 T + 26030 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 186708 T^{2} + 13784867446 T^{4} - 186708 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 436 T + 102166 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 57900 T^{2} + 11793718966 T^{4} - 57900 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 224 T + 179422 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 442740 T^{2} + 87795274358 T^{4} - 442740 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 305782 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 348986 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 140620 T^{2} + 86210675286 T^{4} - 140620 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 2048 T + 1763566 T^{2} + 2048 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 660 T + 407702 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 496 T + 323534 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1659916 T^{2} + 1244512983830 T^{4} - 1659916 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 532 T + 1467382 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1220 T + 586694 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.59207575401755753631206101219, −6.26062472792365490459420776739, −6.24255281704662336077519130889, −5.85170424767434661798252148823, −5.79258111050766000520836999305, −5.56436358646384592628065864021, −5.22303203853621308770461437635, −4.85766238353736897406402653782, −4.85579691800265619490730407924, −4.61387043936934815483749938222, −4.39363523035148622953374001041, −3.98968452482039685194856849941, −3.93502296422732894008056490511, −3.58050804708095163974975248100, −3.41351867600392993308045333719, −2.84166766365816001793022049532, −2.71006937492885790873837198597, −2.69225880764399891942690258286, −1.89614750482167420165937717160, −1.73083344713812760762707842141, −1.55926159552243399233309041027, −1.48073275965442814629065039252, −1.22471915472832010392272064191, −0.29373153488863885471335450159, −0.06550150401118963810054651172,
0.06550150401118963810054651172, 0.29373153488863885471335450159, 1.22471915472832010392272064191, 1.48073275965442814629065039252, 1.55926159552243399233309041027, 1.73083344713812760762707842141, 1.89614750482167420165937717160, 2.69225880764399891942690258286, 2.71006937492885790873837198597, 2.84166766365816001793022049532, 3.41351867600392993308045333719, 3.58050804708095163974975248100, 3.93502296422732894008056490511, 3.98968452482039685194856849941, 4.39363523035148622953374001041, 4.61387043936934815483749938222, 4.85579691800265619490730407924, 4.85766238353736897406402653782, 5.22303203853621308770461437635, 5.56436358646384592628065864021, 5.79258111050766000520836999305, 5.85170424767434661798252148823, 6.24255281704662336077519130889, 6.26062472792365490459420776739, 6.59207575401755753631206101219
