| L(s) = 1 | − 3-s + 7-s − 2·9-s + 5·11-s + 8·19-s − 21-s − 12·23-s − 3·27-s − 4·29-s + 6·31-s − 5·33-s − 3·37-s − 41-s − 10·43-s − 3·47-s − 14·49-s − 11·53-s − 8·57-s − 4·59-s − 10·61-s − 2·63-s − 28·67-s + 12·69-s + 15·71-s − 5·73-s + 5·77-s + 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.50·11-s + 1.83·19-s − 0.218·21-s − 2.50·23-s − 0.577·27-s − 0.742·29-s + 1.07·31-s − 0.870·33-s − 0.493·37-s − 0.156·41-s − 1.52·43-s − 0.437·47-s − 2·49-s − 1.51·53-s − 1.05·57-s − 0.520·59-s − 1.28·61-s − 0.251·63-s − 3.42·67-s + 1.44·69-s + 1.78·71-s − 0.585·73-s + 0.569·77-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - T + 15 T^{2} - 16 T^{3} + 15 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 3 p T^{2} - 102 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 59 T^{2} - 240 T^{3} + 59 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 59 T^{2} + 200 T^{3} + 59 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 95 T^{2} - 368 T^{3} + 95 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 99 T^{2} + 102 T^{3} + 99 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 3 p T^{2} + 796 T^{3} + 3 p^{2} T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 3 T + 23 T^{2} + 416 T^{3} + 23 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 11 T + 127 T^{2} + 714 T^{3} + 127 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 115 T^{2} + 240 T^{3} + 115 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 115 T^{2} + 1180 T^{3} + 115 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 28 T + 395 T^{2} + 3792 T^{3} + 395 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 15 T + 245 T^{2} - 2098 T^{3} + 245 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 5 T + 99 T^{2} - 34 T^{3} + 99 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 143 T^{2} - 528 T^{3} + 143 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 5 T + 19 T^{2} + 936 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 119 T^{2} + 1268 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
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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
−7.51324853141664989075115556771, −6.83358317106902259988308181339, −6.82873956341494634215173442120, −6.70981827896941624388544984637, −6.28710439148902304661632405925, −6.13057589739697793638175556107, −6.05213078927536172290000804079, −5.58140110804106191211953755825, −5.47073697325298995107249971673, −5.44576394383460492769191194223, −4.82605194784495360111752779783, −4.78882128701540753907897466860, −4.44319310702920778826397828927, −4.33684346644186151302207492599, −3.84764657726740936294703733199, −3.73251354655161926106307524404, −3.30326041543959667181872775495, −3.27477316409797260680930093607, −3.01920208306863777911728063995, −2.47058707353517914456244618904, −2.28567946954683822547499838890, −1.76022000084607787165277672748, −1.52320953841679416450685071918, −1.25519533529554292368778870602, −1.24053034926317289187995904282, 0, 0, 0,
1.24053034926317289187995904282, 1.25519533529554292368778870602, 1.52320953841679416450685071918, 1.76022000084607787165277672748, 2.28567946954683822547499838890, 2.47058707353517914456244618904, 3.01920208306863777911728063995, 3.27477316409797260680930093607, 3.30326041543959667181872775495, 3.73251354655161926106307524404, 3.84764657726740936294703733199, 4.33684346644186151302207492599, 4.44319310702920778826397828927, 4.78882128701540753907897466860, 4.82605194784495360111752779783, 5.44576394383460492769191194223, 5.47073697325298995107249971673, 5.58140110804106191211953755825, 6.05213078927536172290000804079, 6.13057589739697793638175556107, 6.28710439148902304661632405925, 6.70981827896941624388544984637, 6.82873956341494634215173442120, 6.83358317106902259988308181339, 7.51324853141664989075115556771
