| L(s) = 1 | − 2·4-s − 2·9-s + 8·11-s + 3·16-s − 10·19-s − 22·31-s + 4·36-s + 8·41-s − 16·44-s + 25·49-s − 40·59-s + 18·61-s − 4·64-s − 32·71-s + 20·76-s + 30·79-s + 3·81-s − 16·99-s + 28·101-s − 10·109-s − 4·121-s + 44·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 2.41·11-s + 3/4·16-s − 2.29·19-s − 3.95·31-s + 2/3·36-s + 1.24·41-s − 2.41·44-s + 25/7·49-s − 5.20·59-s + 2.30·61-s − 1/2·64-s − 3.79·71-s + 2.29·76-s + 3.37·79-s + 1/3·81-s − 1.60·99-s + 2.78·101-s − 0.957·109-s − 0.363·121-s + 3.95·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.465332636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.465332636\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 253 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 833 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 798 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4$ | \( ( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 4513 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 25 T^{2} - 1427 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5238 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 35 T^{2} + 9133 T^{4} + 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 265 T^{2} + 28113 T^{4} - 265 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 15 T + 153 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 18558 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 305 T^{2} + 40713 T^{4} - 305 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.92541989240301538922892186796, −5.87956153161814150806247218790, −5.62788673333835665270908712271, −5.59203534736063528824747659795, −5.16046187308634737555539735552, −5.11159272234698165003434211089, −4.64170257791474175995932618471, −4.48110060224336278130074818253, −4.43884386664024771932038937377, −4.19329490226105051926147348167, −4.04589909179973170841512493606, −3.76646766970963308506877859164, −3.59918743409486049649947372536, −3.45918963316304720752824956366, −3.31273624196281177328862742935, −2.82044149890766579348615714802, −2.68832708830783905300532935973, −2.27568611279244327934031872991, −2.05942701582556055700100786578, −1.84824003597774527765388113141, −1.67260158097575729250416718126, −1.21859972340209679448805039412, −1.05962577466257744471066788801, −0.47707469949786367682616603827, −0.26137412060116223779126428110,
0.26137412060116223779126428110, 0.47707469949786367682616603827, 1.05962577466257744471066788801, 1.21859972340209679448805039412, 1.67260158097575729250416718126, 1.84824003597774527765388113141, 2.05942701582556055700100786578, 2.27568611279244327934031872991, 2.68832708830783905300532935973, 2.82044149890766579348615714802, 3.31273624196281177328862742935, 3.45918963316304720752824956366, 3.59918743409486049649947372536, 3.76646766970963308506877859164, 4.04589909179973170841512493606, 4.19329490226105051926147348167, 4.43884386664024771932038937377, 4.48110060224336278130074818253, 4.64170257791474175995932618471, 5.11159272234698165003434211089, 5.16046187308634737555539735552, 5.59203534736063528824747659795, 5.62788673333835665270908712271, 5.87956153161814150806247218790, 5.92541989240301538922892186796
