| L(s) = 1 | + 3·3-s − 4-s + 3·8-s + 2·9-s − 8·11-s − 3·12-s − 5·13-s − 16-s − 7·17-s − 13·19-s − 23-s + 9·24-s − 15·25-s − 4·27-s + 2·31-s − 6·32-s − 24·33-s − 2·36-s − 13·37-s − 15·39-s + 3·41-s − 43-s + 8·44-s + 9·47-s − 3·48-s − 21·51-s + 5·52-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1/2·4-s + 1.06·8-s + 2/3·9-s − 2.41·11-s − 0.866·12-s − 1.38·13-s − 1/4·16-s − 1.69·17-s − 2.98·19-s − 0.208·23-s + 1.83·24-s − 3·25-s − 0.769·27-s + 0.359·31-s − 1.06·32-s − 4.17·33-s − 1/3·36-s − 2.13·37-s − 2.40·39-s + 0.468·41-s − 0.152·43-s + 1.20·44-s + 1.31·47-s − 0.433·48-s − 2.94·51-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T^{2} - 3 T^{3} + p T^{4} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - p T + 7 T^{2} - 11 T^{3} + 7 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 37 T^{2} + 136 T^{3} + 37 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 21 T^{2} + 67 T^{3} + 21 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + 59 T^{2} + 231 T^{3} + 59 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 13 T + 103 T^{2} + 543 T^{3} + 103 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 49 T^{2} + 71 T^{3} + 49 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 67 T^{2} - 24 T^{3} + 67 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 25 T^{2} - 180 T^{3} + 25 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 13 T + 141 T^{2} + 953 T^{3} + 141 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + T + 121 T^{2} + 83 T^{3} + 121 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 129 T^{2} - 713 T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 163 T^{2} + 808 T^{3} + 163 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 85 T^{2} - 640 T^{3} + 85 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T - 17 T^{2} + 444 T^{3} - 17 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 125 T^{2} - 8 p T^{3} + 125 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 205 T^{2} + 1220 T^{3} + 205 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 147 T^{2} - 956 T^{3} + 147 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 281 T^{2} + 2268 T^{3} + 281 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 5 T + 249 T^{2} + 827 T^{3} + 249 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 19 T + 387 T^{2} + 3749 T^{3} + 387 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.630340393445771560208434823157, −8.292483788187114797969082005730, −8.088172379311099567317188936908, −7.65813891069424041270469568620, −7.60153102117575966340901271039, −7.55558071237669283913842001020, −6.97047929102979903919615334475, −6.81904310227978894492038661843, −6.48689582973945590162471226586, −6.16486887556350090394728204908, −5.66054099671768193483343218443, −5.53427996996419030407718575318, −5.23346997891074903716521334247, −4.94952083500728255524984265889, −4.44003912332878462130565902485, −4.40871969089400311183891797964, −4.10720660670331164054673183032, −3.82739016922514213309924166526, −3.54845148016178123128373556278, −2.78306489845453934112263348417, −2.71343109072403324819307531405, −2.35091305718228650081320093736, −2.27594171943160421296334322268, −1.85234514368544604899152210122, −1.64674441952210961747138984810, 0, 0, 0,
1.64674441952210961747138984810, 1.85234514368544604899152210122, 2.27594171943160421296334322268, 2.35091305718228650081320093736, 2.71343109072403324819307531405, 2.78306489845453934112263348417, 3.54845148016178123128373556278, 3.82739016922514213309924166526, 4.10720660670331164054673183032, 4.40871969089400311183891797964, 4.44003912332878462130565902485, 4.94952083500728255524984265889, 5.23346997891074903716521334247, 5.53427996996419030407718575318, 5.66054099671768193483343218443, 6.16486887556350090394728204908, 6.48689582973945590162471226586, 6.81904310227978894492038661843, 6.97047929102979903919615334475, 7.55558071237669283913842001020, 7.60153102117575966340901271039, 7.65813891069424041270469568620, 8.088172379311099567317188936908, 8.292483788187114797969082005730, 8.630340393445771560208434823157
