| L(s) = 1 | − 2·2-s − 8·3-s − 4-s − 10·5-s + 16·6-s − 8·7-s − 4·8-s + 6·9-s + 20·10-s + 8·12-s + 130·13-s + 16·14-s + 80·15-s − 19·16-s − 14·17-s − 12·18-s − 48·19-s + 10·20-s + 64·21-s − 128·23-s + 32·24-s − 163·25-s − 260·26-s + 200·27-s + 8·28-s − 30·29-s − 160·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.53·3-s − 1/8·4-s − 0.894·5-s + 1.08·6-s − 0.431·7-s − 0.176·8-s + 2/9·9-s + 0.632·10-s + 0.192·12-s + 2.77·13-s + 0.305·14-s + 1.37·15-s − 0.296·16-s − 0.199·17-s − 0.157·18-s − 0.579·19-s + 0.111·20-s + 0.665·21-s − 1.16·23-s + 0.272·24-s − 1.30·25-s − 1.96·26-s + 1.42·27-s + 0.0539·28-s − 0.192·29-s − 0.973·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 5 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 p T + 263 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 p T + 8607 T^{2} - 10 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 14 T + 5987 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 48 T + 13322 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 128 T + 15362 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 30 T + 32575 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 184 T + 41538 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 126 T + 30383 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 370 T + 172019 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 264 T + 170630 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 256 T + 76178 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 162 T + 217615 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1304 T + 814694 T^{2} + 1304 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 300 T + 374894 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 656 T + 707658 T^{2} + 656 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1176 T + 1023934 T^{2} + 1176 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 668 T + 500790 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 416 T + 886770 T^{2} - 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 960 T + 1342762 T^{2} + 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1074 T + 1326595 T^{2} + 1074 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 338 T + 1196835 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.54312902527990078731357699332, −11.84136519494793570888178182090, −11.69749867106793630017360748346, −11.13932107379025067514809306668, −10.77929667270135587198217964109, −10.40649237575135830075911860622, −9.256919597369267883943543751783, −9.199897242164761889585151617176, −8.247009564017881153260004804128, −8.163654637747614970603094044866, −7.21328586279761425643477272076, −6.26127963803604734557369248215, −5.93378270541447540357933678623, −5.83909254448577728347130637957, −4.41226964582037130872484182666, −4.00707281190307933309283811031, −3.09168785190754606331168213997, −1.42874796647025140160111361424, 0, 0,
1.42874796647025140160111361424, 3.09168785190754606331168213997, 4.00707281190307933309283811031, 4.41226964582037130872484182666, 5.83909254448577728347130637957, 5.93378270541447540357933678623, 6.26127963803604734557369248215, 7.21328586279761425643477272076, 8.163654637747614970603094044866, 8.247009564017881153260004804128, 9.199897242164761889585151617176, 9.256919597369267883943543751783, 10.40649237575135830075911860622, 10.77929667270135587198217964109, 11.13932107379025067514809306668, 11.69749867106793630017360748346, 11.84136519494793570888178182090, 12.54312902527990078731357699332
