| L(s) = 1 | − 24·13-s + 24·19-s + 68·25-s + 152·31-s + 128·37-s + 80·43-s + 14·49-s + 48·61-s − 24·67-s + 16·73-s + 88·79-s + 160·97-s − 440·103-s + 176·109-s + 272·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 92·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.84·13-s + 1.26·19-s + 2.71·25-s + 4.90·31-s + 3.45·37-s + 1.86·43-s + 2/7·49-s + 0.786·61-s − 0.358·67-s + 0.219·73-s + 1.11·79-s + 1.64·97-s − 4.27·103-s + 1.61·109-s + 2.24·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.544·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.540186425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.540186425\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2294 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 36578 T^{4} - 272 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 12 T + 262 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 59754 T^{4} - 4 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 12 T + 58 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 784 T^{2} + 386754 T^{4} - 784 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1136 T^{2} + 1500194 T^{4} - 1136 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 76 T + 3338 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 64 T + 3650 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2660 T^{2} + 5130134 T^{4} - 2660 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 40 T + 3090 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7252 T^{2} + 22326630 T^{4} - 7252 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 7168 T^{2} + 28480866 T^{4} - 7168 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7732 T^{2} + 33955206 T^{4} - 7732 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 24 T - 2522 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 12 T + 3526 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 14800 T^{2} + 101192514 T^{4} - 14800 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 9302 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 26404 T^{2} + 269064294 T^{4} - 26404 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14084 T^{2} + 163244246 T^{4} - 14084 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 80 T + 20390 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−5.86641259923729594743906480061, −5.68725479710315349387034015196, −5.21884555757578393775346400685, −5.20216726466538071705756834330, −4.87864453018403088189042225893, −4.75262923109394422051306492429, −4.55494525352077223599875776020, −4.54704005535664860796434472662, −4.38644696035700673996053440969, −4.04023298827181804042241566821, −3.76983788250128699040913724074, −3.48621379354279414923095376171, −3.24322313981189872417949907156, −3.01877929283486309235246786872, −2.68764197833032863498126304955, −2.62019739966944000376357441349, −2.54792858560335322502030681779, −2.29368643013511108579923156203, −2.24148603837782397140068754634, −1.35902345943614004431466463508, −1.27679417302914843723682111966, −1.06596873266656261010867370077, −0.794358690215971988726351645388, −0.69353650909166399234193146870, −0.31146552924019540001097714529,
0.31146552924019540001097714529, 0.69353650909166399234193146870, 0.794358690215971988726351645388, 1.06596873266656261010867370077, 1.27679417302914843723682111966, 1.35902345943614004431466463508, 2.24148603837782397140068754634, 2.29368643013511108579923156203, 2.54792858560335322502030681779, 2.62019739966944000376357441349, 2.68764197833032863498126304955, 3.01877929283486309235246786872, 3.24322313981189872417949907156, 3.48621379354279414923095376171, 3.76983788250128699040913724074, 4.04023298827181804042241566821, 4.38644696035700673996053440969, 4.54704005535664860796434472662, 4.55494525352077223599875776020, 4.75262923109394422051306492429, 4.87864453018403088189042225893, 5.20216726466538071705756834330, 5.21884555757578393775346400685, 5.68725479710315349387034015196, 5.86641259923729594743906480061
