| L(s) = 1 | − 12·11-s + 12·19-s − 8·25-s + 12·43-s − 12·49-s − 36·59-s + 4·67-s − 24·73-s − 12·83-s + 24·89-s − 12·107-s + 24·113-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.61·11-s + 2.75·19-s − 8/5·25-s + 1.82·43-s − 1.71·49-s − 4.68·59-s + 0.488·67-s − 2.80·73-s − 1.31·83-s + 2.54·89-s − 1.16·107-s + 2.25·113-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 12 T^{2} + 86 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 24 T^{2} + 290 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 1110 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 120 T^{2} + 6146 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 4674 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 216 T^{2} + 18914 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 236 T^{2} + 23574 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 60 T^{2} + 1094 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 182 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.76586023163380844735023261844, −5.56305602401833478554533887826, −5.41340794334289390807313225086, −5.17083454201060076641059943229, −5.06584579977950825697189783505, −4.84483218690414016283951175678, −4.75691052501699095455252087629, −4.60656649933578785513102597679, −4.57561431435825007113081002319, −4.00795793533278256012302000519, −3.81019954474159096782987636640, −3.79026564454694821170095185796, −3.55897407453321397963713090078, −3.17829072932933740725510268348, −3.01362713234189801599954230886, −2.97866686790889399500362693118, −2.88522781551283893534036070942, −2.44171206121185618103970598907, −2.38717062268212557857373040493, −2.14635524754993739289785962849, −2.06819093831957746679997156532, −1.44465008667432960132642291136, −1.22020415911959115229984470536, −1.20671576777011779905573360833, −1.05140726223231886858073163856, 0, 0, 0, 0,
1.05140726223231886858073163856, 1.20671576777011779905573360833, 1.22020415911959115229984470536, 1.44465008667432960132642291136, 2.06819093831957746679997156532, 2.14635524754993739289785962849, 2.38717062268212557857373040493, 2.44171206121185618103970598907, 2.88522781551283893534036070942, 2.97866686790889399500362693118, 3.01362713234189801599954230886, 3.17829072932933740725510268348, 3.55897407453321397963713090078, 3.79026564454694821170095185796, 3.81019954474159096782987636640, 4.00795793533278256012302000519, 4.57561431435825007113081002319, 4.60656649933578785513102597679, 4.75691052501699095455252087629, 4.84483218690414016283951175678, 5.06584579977950825697189783505, 5.17083454201060076641059943229, 5.41340794334289390807313225086, 5.56305602401833478554533887826, 5.76586023163380844735023261844
