| L(s) = 1 | − 4-s − 18·9-s − 44·11-s − 111·16-s − 204·19-s + 392·29-s + 300·31-s + 18·36-s − 352·41-s + 44·44-s − 98·49-s + 1.68e3·59-s − 408·61-s + 159·64-s + 3.34e3·71-s + 204·76-s + 1.77e3·79-s + 243·81-s − 1.17e3·89-s + 792·99-s − 64·101-s + 608·109-s − 392·116-s − 3.98e3·121-s − 300·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/8·4-s − 2/3·9-s − 1.20·11-s − 1.73·16-s − 2.46·19-s + 2.51·29-s + 1.73·31-s + 1/12·36-s − 1.34·41-s + 0.150·44-s − 2/7·49-s + 3.72·59-s − 0.856·61-s + 0.310·64-s + 5.58·71-s + 0.307·76-s + 2.52·79-s + 1/3·81-s − 1.40·89-s + 0.804·99-s − 0.0630·101-s + 0.534·109-s − 0.313·116-s − 2.99·121-s − 0.217·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1923324772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1923324772\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} + 7 p^{4} T^{4} + p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 p T + 2718 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8416 T^{2} + 27329422 T^{4} - 8416 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4604 T^{2} - 2402618 T^{4} - 4604 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 102 T + 13134 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1868 T^{2} - 142455866 T^{4} - 1868 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 196 T + 20942 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 150 T + 36542 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 155884 T^{2} + 11012319222 T^{4} - 155884 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 176 T - 16914 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 225580 T^{2} + 23395199158 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 183612 T^{2} + 18245588294 T^{4} - 183612 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 302000 T^{2} + 54356941198 T^{4} - 302000 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 844 T + 474182 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 204 T + 455006 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1064524 T^{2} + 463499688022 T^{4} - 1064524 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1670 T + 1384382 T^{2} - 1670 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 155440 T^{2} + 209914818238 T^{4} - 155440 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 888 T + 1007454 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 340236 T^{2} + 29905619222 T^{4} - 340236 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 588 T + 1495334 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 310080 T^{2} + 1253411633918 T^{4} - 310080 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−7.46468960638725468710410316764, −6.85758866460490217686195210065, −6.81884674080411514778326676708, −6.70563780339687115985473973071, −6.45658335329951133032491410051, −6.31714700248074349796356714708, −6.21815021159355175710758615853, −5.47646647078565872685525504207, −5.24789212667538642918090425330, −5.15095990825133038978404529576, −5.12339337988887692799664641898, −4.52290368930383097628961529034, −4.33430164799673217844166853276, −4.31358716731988401315353683963, −3.65464113277708188772499867254, −3.64905251990573294445280500922, −3.16997452289445418459853478361, −2.62292098925399939831972819075, −2.37717022641922166950124653938, −2.34224726254288258447271356778, −2.25852779927536283359697924654, −1.50147145185159353282173626839, −0.868343270078126560103328904762, −0.69665451496178968429292956999, −0.07812670829720184653250556573,
0.07812670829720184653250556573, 0.69665451496178968429292956999, 0.868343270078126560103328904762, 1.50147145185159353282173626839, 2.25852779927536283359697924654, 2.34224726254288258447271356778, 2.37717022641922166950124653938, 2.62292098925399939831972819075, 3.16997452289445418459853478361, 3.64905251990573294445280500922, 3.65464113277708188772499867254, 4.31358716731988401315353683963, 4.33430164799673217844166853276, 4.52290368930383097628961529034, 5.12339337988887692799664641898, 5.15095990825133038978404529576, 5.24789212667538642918090425330, 5.47646647078565872685525504207, 6.21815021159355175710758615853, 6.31714700248074349796356714708, 6.45658335329951133032491410051, 6.70563780339687115985473973071, 6.81884674080411514778326676708, 6.85758866460490217686195210065, 7.46468960638725468710410316764
