| L(s) = 1 | − 24·7-s − 24·11-s − 48·13-s + 24·17-s − 24·23-s + 20·31-s − 48·37-s + 192·41-s + 72·43-s + 144·47-s + 288·49-s + 120·53-s + 44·61-s − 24·67-s + 96·71-s − 48·73-s + 576·77-s − 9·81-s + 48·83-s + 1.15e3·91-s + 192·97-s + 240·101-s − 48·103-s − 360·107-s − 288·113-s − 576·119-s + 308·121-s + ⋯ |
| L(s) = 1 | − 3.42·7-s − 2.18·11-s − 3.69·13-s + 1.41·17-s − 1.04·23-s + 0.645·31-s − 1.29·37-s + 4.68·41-s + 1.67·43-s + 3.06·47-s + 5.87·49-s + 2.26·53-s + 0.721·61-s − 0.358·67-s + 1.35·71-s − 0.657·73-s + 7.48·77-s − 1/9·81-s + 0.578·83-s + 12.6·91-s + 1.97·97-s + 2.37·101-s − 0.466·103-s − 3.36·107-s − 2.54·113-s − 4.84·119-s + 2.54·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.221471350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221471350\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2832 T^{3} + 23087 T^{4} + 2832 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 12 T + 62 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 20640 T^{3} + 301679 T^{4} + 20640 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 6072 T^{3} + 126722 T^{4} - 6072 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 290 T^{2} - 30237 T^{4} - 290 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} - 8904 T^{3} - 534718 T^{4} - 8904 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2860 T^{2} + 3428358 T^{4} - 2860 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 10 T + 3 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 21936 T^{3} - 414046 T^{4} + 21936 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 96 T + 5450 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 72 T + 2592 T^{2} - 174384 T^{3} + 11403839 T^{4} - 174384 p^{2} T^{5} + 2592 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 144 T + 10368 T^{2} - 551376 T^{3} + 26698082 T^{4} - 551376 p^{2} T^{5} + 10368 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 120 T + 7200 T^{2} - 345720 T^{3} + 16595138 T^{4} - 345720 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6062 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 22 T + 4107 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 103632 T^{3} + 37261007 T^{4} + 103632 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 48 T + 10442 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 156720 T^{3} + 17060354 T^{4} + 156720 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 23228 T^{2} + 212771334 T^{4} - 23228 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 90480 T^{3} - 17933566 T^{4} - 90480 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 29668 T^{2} + 345034374 T^{4} - 29668 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 192 T + 18432 T^{2} - 2676864 T^{3} + 368210639 T^{4} - 2676864 p^{2} T^{5} + 18432 p^{4} T^{6} - 192 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.96646308043154752181297970304, −6.52248285267377988059035785870, −6.17905373409378869707533036858, −6.12169122039794265539450500673, −5.87679886585109379021266805905, −5.72905916004528641414694977266, −5.31908132585292989470572603516, −5.26059200042317281586202758491, −5.25449563931442851387291846628, −4.74423330841622047944221081317, −4.41435994892506342405786324241, −4.08503424591572153689399543148, −4.03658142746324149281189152541, −3.79801915471955881209178657080, −3.39342719905668860747228958937, −3.01299649611871756992015656577, −2.88301508453410255445868069828, −2.56466016272689648545563954618, −2.38898855904227218528840511692, −2.34404009596390475229535528528, −2.29796808852470786921156059636, −1.20226999584431711328225015330, −0.63438009590278729606838446092, −0.49531217436709985585783240521, −0.34531496486847060580837417062,
0.34531496486847060580837417062, 0.49531217436709985585783240521, 0.63438009590278729606838446092, 1.20226999584431711328225015330, 2.29796808852470786921156059636, 2.34404009596390475229535528528, 2.38898855904227218528840511692, 2.56466016272689648545563954618, 2.88301508453410255445868069828, 3.01299649611871756992015656577, 3.39342719905668860747228958937, 3.79801915471955881209178657080, 4.03658142746324149281189152541, 4.08503424591572153689399543148, 4.41435994892506342405786324241, 4.74423330841622047944221081317, 5.25449563931442851387291846628, 5.26059200042317281586202758491, 5.31908132585292989470572603516, 5.72905916004528641414694977266, 5.87679886585109379021266805905, 6.12169122039794265539450500673, 6.17905373409378869707533036858, 6.52248285267377988059035785870, 6.96646308043154752181297970304
