| L(s) = 1 | − 2-s − 6·3-s − 4·5-s + 6·6-s − 10·7-s + 23·9-s + 4·10-s − 4·11-s − 3·13-s + 10·14-s + 24·15-s − 6·17-s − 23·18-s − 2·19-s + 60·21-s + 4·22-s + 6·23-s + 5·25-s + 3·26-s − 70·27-s − 16·29-s − 24·30-s − 4·31-s + 32-s + 24·33-s + 6·34-s + 40·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3.46·3-s − 1.78·5-s + 2.44·6-s − 3.77·7-s + 23/3·9-s + 1.26·10-s − 1.20·11-s − 0.832·13-s + 2.67·14-s + 6.19·15-s − 1.45·17-s − 5.42·18-s − 0.458·19-s + 13.0·21-s + 0.852·22-s + 1.25·23-s + 25-s + 0.588·26-s − 13.4·27-s − 2.97·29-s − 4.38·30-s − 0.718·31-s + 0.176·32-s + 4.17·33-s + 1.02·34-s + 6.76·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 + 2 p T + 13 T^{2} + 10 T^{3} + T^{4} + 10 p T^{5} + 13 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 4 T + 11 T^{2} + 34 T^{3} + 101 T^{4} + 34 p T^{5} + 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 10 T + 53 T^{2} + 200 T^{3} + 589 T^{4} + 200 p T^{5} + 53 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 3 T - 9 T^{2} - 11 T^{3} + 144 T^{4} - 11 p T^{5} - 9 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 6 T + 19 T^{2} + 132 T^{3} + 829 T^{4} + 132 p T^{5} + 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + 2 T - 15 T^{2} - 68 T^{3} + 149 T^{4} - 68 p T^{5} - 15 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 16 T + 67 T^{2} - 412 T^{3} - 4935 T^{4} - 412 p T^{5} + 67 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 4 T - 25 T^{2} - 94 T^{3} + 559 T^{4} - 94 p T^{5} - 25 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 8 T + 27 T^{2} + 80 T^{3} - 1639 T^{4} + 80 p T^{5} + 27 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 6 T + 35 T^{2} + 384 T^{3} + 3829 T^{4} + 384 p T^{5} + 35 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 3 T + 7 T^{2} - 315 T^{3} + 3136 T^{4} - 315 p T^{5} + 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 26 T + 403 T^{2} + 4480 T^{3} + 37041 T^{4} + 4480 p T^{5} + 403 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 14 T + 137 T^{2} + 1442 T^{3} + 14555 T^{4} + 1442 p T^{5} + 137 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 6 T + 15 T^{2} + 544 T^{3} + 6909 T^{4} + 544 p T^{5} + 15 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 7 T + 85 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4} )( 1 + 31 T + 381 T^{2} + 31 p T^{3} + p^{2} T^{4} ) \) |
| 73 | $C_2^2:C_4$ | \( 1 + 4 T + 23 T^{2} + 620 T^{3} + 7921 T^{4} + 620 p T^{5} + 23 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 16 T + 27 T^{2} - 382 T^{3} - 745 T^{4} - 382 p T^{5} + 27 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 4 T - 77 T^{2} - 250 T^{3} + 5811 T^{4} - 250 p T^{5} - 77 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 182 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 7 T + 27 T^{2} - 1085 T^{3} + 16976 T^{4} - 1085 p T^{5} + 27 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.60600626271005327668954865458, −7.35859493207165480161557789088, −7.05247064691803186037805555275, −7.05240566361550233550862358251, −6.51702503707571565073848400937, −6.49505523125970996289368462650, −6.44583759545739627498228250901, −6.15542419436794388353825044512, −5.88489411055834332082721236680, −5.73243968216894180924080341266, −5.53409810839927335619512063614, −5.11868973736047526495140721090, −4.87801691201674992465696504668, −4.69509491853901992389701466150, −4.34704124494827544355028450274, −4.28215682221501953957662102990, −4.07437764627496938101807575434, −3.43327400204107589288285683209, −3.42649616495290332098348432828, −3.21855137688620573861877135235, −2.90573280042914238529022317528, −2.38666710461711832904972469695, −1.75244250495356496501454192637, −1.61591702770766271063553821413, −0.76859286835796455205723676878, 0, 0, 0, 0,
0.76859286835796455205723676878, 1.61591702770766271063553821413, 1.75244250495356496501454192637, 2.38666710461711832904972469695, 2.90573280042914238529022317528, 3.21855137688620573861877135235, 3.42649616495290332098348432828, 3.43327400204107589288285683209, 4.07437764627496938101807575434, 4.28215682221501953957662102990, 4.34704124494827544355028450274, 4.69509491853901992389701466150, 4.87801691201674992465696504668, 5.11868973736047526495140721090, 5.53409810839927335619512063614, 5.73243968216894180924080341266, 5.88489411055834332082721236680, 6.15542419436794388353825044512, 6.44583759545739627498228250901, 6.49505523125970996289368462650, 6.51702503707571565073848400937, 7.05240566361550233550862358251, 7.05247064691803186037805555275, 7.35859493207165480161557789088, 7.60600626271005327668954865458
