| L(s) = 1 | − 2-s − 3·5-s + 5·7-s + 3·10-s + 2·11-s + 4·13-s − 5·14-s − 16-s + 2·17-s + 4·19-s − 2·22-s − 3·23-s + 6·25-s − 4·26-s + 7·29-s + 8·31-s − 32-s − 2·34-s − 15·35-s + 6·37-s − 4·38-s + 13·41-s + 10·43-s + 3·46-s − 13·47-s + 49-s − 6·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s + 1.88·7-s + 0.948·10-s + 0.603·11-s + 1.10·13-s − 1.33·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.426·22-s − 0.625·23-s + 6/5·25-s − 0.784·26-s + 1.29·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s − 2.53·35-s + 0.986·37-s − 0.648·38-s + 2.03·41-s + 1.52·43-s + 0.442·46-s − 1.89·47-s + 1/7·49-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66430125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551401020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551401020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5 T + 24 T^{2} - 67 T^{3} + 24 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 25 T^{2} - 32 T^{3} + 25 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 35 T^{2} - 100 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 43 T^{2} - 56 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 148 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 36 T^{2} + 21 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 2 p T^{2} - 355 T^{3} + 2 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 33 T^{2} - 28 T^{3} + 33 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 99 T^{2} - 440 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 13 T + 142 T^{2} - 1069 T^{3} + 142 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 125 T^{2} - 856 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 13 T + 130 T^{2} + 853 T^{3} + 130 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 139 T^{2} - 188 T^{3} + 139 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 157 T^{2} - 212 T^{3} + 157 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 146 T^{2} - 193 T^{3} + 146 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 11 T + 162 T^{2} - 967 T^{3} + 162 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 121 T^{2} - 712 T^{3} + 121 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 8 T + 155 T^{2} + 1040 T^{3} + 155 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 153 T^{2} - 340 T^{3} + 153 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 15 T + 276 T^{2} - 2409 T^{3} + 276 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 + 18 T + 255 T^{2} + 2188 T^{3} + 255 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07146441959455442372232616676, −9.476007814914580912086630166011, −9.421856993621271807439932325262, −9.306546041442950052020454251449, −8.533713877497509955171401547219, −8.397483001033902170637563682929, −8.244814752793429897264647805926, −7.939104289954117485209762841929, −7.79205900265118743413620976189, −7.44243333622413619716615056519, −6.91898632223280747240435368876, −6.66347051677547516233021240332, −6.17699643520193400866159855186, −5.98567514029642307861749193439, −5.22666800839247342637974200460, −5.21494526850361755671440842486, −4.46965434298240444207065051840, −4.39699754656650699664459128108, −4.11886575434530500089847515341, −3.49752636174282218812187065946, −3.13793794920203995567181359917, −2.55449225267992608826531435509, −1.89342420448833102084428967755, −1.05768489668298083823781522462, −0.954570366014384757362295281472,
0.954570366014384757362295281472, 1.05768489668298083823781522462, 1.89342420448833102084428967755, 2.55449225267992608826531435509, 3.13793794920203995567181359917, 3.49752636174282218812187065946, 4.11886575434530500089847515341, 4.39699754656650699664459128108, 4.46965434298240444207065051840, 5.21494526850361755671440842486, 5.22666800839247342637974200460, 5.98567514029642307861749193439, 6.17699643520193400866159855186, 6.66347051677547516233021240332, 6.91898632223280747240435368876, 7.44243333622413619716615056519, 7.79205900265118743413620976189, 7.939104289954117485209762841929, 8.244814752793429897264647805926, 8.397483001033902170637563682929, 8.533713877497509955171401547219, 9.306546041442950052020454251449, 9.421856993621271807439932325262, 9.476007814914580912086630166011, 10.07146441959455442372232616676
