| L(s) = 1 | − 2·5-s − 4·11-s + 3·13-s − 6·17-s + 25-s − 6·29-s − 8·31-s + 2·37-s − 10·41-s + 4·43-s + 8·47-s − 5·49-s − 14·53-s + 8·55-s − 20·59-s + 2·61-s − 6·65-s − 16·67-s + 8·71-s + 22·73-s − 36·79-s − 4·83-s + 12·85-s + 6·89-s − 2·97-s − 6·101-s − 28·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.20·11-s + 0.832·13-s − 1.45·17-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.328·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 5/7·49-s − 1.92·53-s + 1.07·55-s − 2.60·59-s + 0.256·61-s − 0.744·65-s − 1.95·67-s + 0.949·71-s + 2.57·73-s − 4.05·79-s − 0.439·83-s + 1.30·85-s + 0.635·89-s − 0.203·97-s − 0.597·101-s − 2.75·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 56 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 41 T^{2} + 16 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 61 T^{2} + 224 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 59 T^{2} - 108 T^{3} + 59 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 143 T^{2} + 812 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 280 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 720 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 171 T^{2} + 1332 T^{3} + 171 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 20 T + 273 T^{2} + 2328 T^{3} + 273 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 16 T + 137 T^{2} + 1104 T^{3} + 137 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 197 T^{2} - 976 T^{3} + 197 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 327 T^{2} - 3220 T^{3} + 327 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 217 T^{2} + 696 T^{3} + 217 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 159 T^{2} - 852 T^{3} + 159 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 239 T^{2} + 348 T^{3} + 239 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.060075035678884879399531429118, −7.61688404820186998899285648138, −7.47557390509396427018589878143, −7.27219446418248418344265826066, −6.82492143040815592550082672363, −6.78041392997042213145000087249, −6.53493475651358488646551380391, −6.09900040728495113540524150884, −5.86237492087646943619293221464, −5.68333500682587525358335729836, −5.46020096559540566522851589011, −5.01832010177739854468275385703, −4.87009035008758904564990681883, −4.56184701465775866478694369938, −4.26169233818429002857627996313, −4.09719233588164838391900696898, −3.67310436181455478635757679017, −3.44076801967697816800379304937, −3.30326384920402681131239776789, −2.75289971560844164960282454744, −2.49631119810651316349746927676, −2.36291370244336333494585776893, −1.58679715639157800024865947374, −1.57938427913110355421802480159, −1.17718036376539871890809236038, 0, 0, 0,
1.17718036376539871890809236038, 1.57938427913110355421802480159, 1.58679715639157800024865947374, 2.36291370244336333494585776893, 2.49631119810651316349746927676, 2.75289971560844164960282454744, 3.30326384920402681131239776789, 3.44076801967697816800379304937, 3.67310436181455478635757679017, 4.09719233588164838391900696898, 4.26169233818429002857627996313, 4.56184701465775866478694369938, 4.87009035008758904564990681883, 5.01832010177739854468275385703, 5.46020096559540566522851589011, 5.68333500682587525358335729836, 5.86237492087646943619293221464, 6.09900040728495113540524150884, 6.53493475651358488646551380391, 6.78041392997042213145000087249, 6.82492143040815592550082672363, 7.27219446418248418344265826066, 7.47557390509396427018589878143, 7.61688404820186998899285648138, 8.060075035678884879399531429118
