| L(s) = 1 | + 22·5-s − 24·7-s + 55·9-s + 194·11-s − 1.07e3·17-s + 654·19-s − 1.09e3·23-s − 1.20e3·25-s − 528·35-s − 4.10e3·43-s + 1.21e3·45-s − 5.25e3·47-s − 5.25e3·49-s + 4.26e3·55-s + 6.07e3·61-s − 1.32e3·63-s − 9.85e3·73-s − 4.65e3·77-s − 1.03e4·81-s − 3.86e3·83-s − 2.36e4·85-s + 1.43e4·95-s + 1.06e4·99-s − 2.39e4·115-s + 2.58e4·119-s − 3.40e4·121-s − 3.25e4·125-s + ⋯ |
| L(s) = 1 | + 0.879·5-s − 0.489·7-s + 0.679·9-s + 1.60·11-s − 3.72·17-s + 1.81·19-s − 2.06·23-s − 1.92·25-s − 0.431·35-s − 2.22·43-s + 0.597·45-s − 2.37·47-s − 2.18·49-s + 1.41·55-s + 1.63·61-s − 0.332·63-s − 1.84·73-s − 0.785·77-s − 1.57·81-s − 0.561·83-s − 3.27·85-s + 1.59·95-s + 1.08·99-s − 1.81·115-s + 1.82·119-s − 2.32·121-s − 2.08·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7585571354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7585571354\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 19 | $D_{4}$ | \( 1 - 654 T + 14206 p T^{2} - 654 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 55 T^{2} + 4460 p T^{4} - 55 p^{8} T^{6} + p^{16} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 11 T + 782 T^{2} - 11 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + 12 T + 2845 T^{2} + 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 97 T + 31136 T^{2} - 97 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 20849 T^{2} + 1540488348 T^{4} + 20849 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 538 T + 207515 T^{2} + 538 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 545 T + 322532 T^{2} + 545 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 2633023 T^{2} + 2733430565076 T^{4} - 2633023 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 2519512 T^{2} + 3273441141390 T^{4} - 2519512 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 7104376 T^{2} + 19605700943118 T^{4} - 7104376 p^{8} T^{6} + p^{16} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 8338264 T^{2} + 31158187139598 T^{4} - 8338264 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2053 T + 7779198 T^{2} + 2053 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 2627 T + 9750236 T^{2} + 2627 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 15479599 T^{2} + 140156197664244 T^{4} - 15479599 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 17627623 T^{2} + 353135309862156 T^{4} - 17627623 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 3039 T + 10663978 T^{2} - 3039 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 50683999 T^{2} + 1452207687034068 T^{4} - 50683999 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 37727824 T^{2} + 1207845843896094 T^{4} - 37727824 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 4928 T + 53379105 T^{2} + 4928 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 49165144 T^{2} + 844121176610958 T^{4} - 49165144 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1934 T + 75521138 T^{2} + 1934 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 59142316 T^{2} + 2302961142454998 T^{4} - 59142316 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 171249712 T^{2} + 17663132345081310 T^{4} - 171249712 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−7.74289353417321726072686916966, −7.74081402269929674819662238214, −7.06296995660080830277299195384, −7.02583772123600969436860681990, −6.63561959456282148604548484281, −6.58780445680302923828940413565, −6.53078015476731659282793417943, −5.84531484949434705201112031895, −5.81129952625154433448013053949, −5.78182392418502112497192382120, −5.09382397700190065327372428318, −4.73451141788453275555540305664, −4.56786324421376733094603909948, −4.33421363621243373435512734249, −4.04539768997432522488062327536, −3.53848982314379997226807555094, −3.48651568414419891908533799216, −3.11579598161688225738112461760, −2.40973846142632962400293178643, −2.26677181021407649872713675519, −1.64671350479105254155941466046, −1.61437807415728380898266585573, −1.57273671101456452442317579990, −0.51375288684573235992880054090, −0.15126052673802571271009999757,
0.15126052673802571271009999757, 0.51375288684573235992880054090, 1.57273671101456452442317579990, 1.61437807415728380898266585573, 1.64671350479105254155941466046, 2.26677181021407649872713675519, 2.40973846142632962400293178643, 3.11579598161688225738112461760, 3.48651568414419891908533799216, 3.53848982314379997226807555094, 4.04539768997432522488062327536, 4.33421363621243373435512734249, 4.56786324421376733094603909948, 4.73451141788453275555540305664, 5.09382397700190065327372428318, 5.78182392418502112497192382120, 5.81129952625154433448013053949, 5.84531484949434705201112031895, 6.53078015476731659282793417943, 6.58780445680302923828940413565, 6.63561959456282148604548484281, 7.02583772123600969436860681990, 7.06296995660080830277299195384, 7.74081402269929674819662238214, 7.74289353417321726072686916966
