| L(s) = 1 | + 976·11-s − 4.16e3·17-s + 1.45e3·19-s + 1.93e4·25-s + 1.17e5·41-s − 1.97e5·43-s + 2.36e5·49-s + 5.42e5·59-s + 7.90e5·67-s + 4.43e5·73-s − 3.46e6·83-s − 7.61e5·89-s − 9.26e5·97-s + 5.79e6·107-s − 2.54e6·113-s − 3.76e6·121-s + ⋯ |
| L(s) = 1 | + 0.733·11-s − 0.848·17-s + 0.212·19-s + 1.23·25-s + 1.71·41-s − 2.48·43-s + 2.00·49-s + 2.63·59-s + 2.62·67-s + 1.14·73-s − 6.05·83-s − 1.07·89-s − 1.01·97-s + 4.73·107-s − 1.76·113-s − 2.12·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(5.063460355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.063460355\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 772 p^{2} T^{2} + 2049246 p^{3} T^{4} - 772 p^{14} T^{6} + p^{24} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 236356 T^{2} + 28021959366 T^{4} - 236356 p^{12} T^{6} + p^{24} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 488 T + 2238078 T^{2} - 488 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 4994596 T^{2} + 30260873415846 T^{4} - 4994596 p^{12} T^{6} + p^{24} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 2084 T + 32560902 T^{2} + 2084 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 728 T + 92449758 T^{2} - 728 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 212117956 T^{2} + 1001159632705962 p T^{4} - 212117956 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1409719204 T^{2} + 1160515330165289766 T^{4} - 1409719204 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 2758360324 T^{2} + 3362667870952277766 T^{4} - 2758360324 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 8121202276 T^{2} + 29600495645847907686 T^{4} - 8121202276 p^{12} T^{6} + p^{24} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 58972 T + 240432438 p T^{2} - 58972 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 2296 p T + 13190101374 T^{2} + 2296 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 13578767236 T^{2} + \)\(16\!\cdots\!86\)\( T^{4} - 13578767236 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 42194545636 T^{2} + \)\(11\!\cdots\!86\)\( T^{4} - 42194545636 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 271016 T + 102678575166 T^{2} - 271016 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 2637195124 p T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - 2637195124 p^{13} T^{6} + p^{24} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 395096 T + 204987839262 T^{2} - 395096 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 164364621124 T^{2} + \)\(21\!\cdots\!06\)\( T^{4} - 164364621124 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 221956 T + 284381984742 T^{2} - 221956 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 828257339524 T^{2} + \)\(28\!\cdots\!06\)\( T^{4} - 828257339524 p^{12} T^{6} + p^{24} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1732504 T + 1400265667422 T^{2} + 1732504 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 380612 T + 970422538278 T^{2} + 380612 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 463388 T + 953987784774 T^{2} + 463388 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55966152506237938023848016718, −7.12829998153716156255133503067, −6.99655210168604422651097698009, −6.74431053293805587827827249863, −6.54830520922983211387925955543, −6.33065599553243754938313119554, −5.83829063821957298764500413176, −5.68484916439075230320847761220, −5.30619687157001469148431330188, −5.16683806506244870592248408079, −4.90669261669277072991367815252, −4.39901808625621284685089517762, −4.17348078199762306577968736533, −3.89332262423651739082680355780, −3.87098845637548448635301540367, −3.21385566316221497743027290598, −3.03881369193657409999809007841, −2.59568833047122304262931973103, −2.32454858319437693379362575975, −2.15172515902447608550998120405, −1.49094076126919347108344329792, −1.30632176824411736691524384835, −0.986859395062464506460861654539, −0.52116450613897478240372024023, −0.31546089857718901907334876481,
0.31546089857718901907334876481, 0.52116450613897478240372024023, 0.986859395062464506460861654539, 1.30632176824411736691524384835, 1.49094076126919347108344329792, 2.15172515902447608550998120405, 2.32454858319437693379362575975, 2.59568833047122304262931973103, 3.03881369193657409999809007841, 3.21385566316221497743027290598, 3.87098845637548448635301540367, 3.89332262423651739082680355780, 4.17348078199762306577968736533, 4.39901808625621284685089517762, 4.90669261669277072991367815252, 5.16683806506244870592248408079, 5.30619687157001469148431330188, 5.68484916439075230320847761220, 5.83829063821957298764500413176, 6.33065599553243754938313119554, 6.54830520922983211387925955543, 6.74431053293805587827827249863, 6.99655210168604422651097698009, 7.12829998153716156255133503067, 7.55966152506237938023848016718
