| L(s) = 1 | − 2·2-s + 8·5-s + 8·8-s − 16·10-s − 44·13-s − 16·16-s − 22·17-s − 2·25-s + 88·26-s + 68·29-s + 44·34-s − 32·37-s + 64·40-s + 26·41-s + 86·49-s + 4·50-s + 104·53-s − 136·58-s − 32·61-s + 64·64-s − 352·65-s − 50·73-s + 64·74-s − 128·80-s − 52·82-s − 176·85-s − 4·89-s + ⋯ |
| L(s) = 1 | − 2-s + 8/5·5-s + 8-s − 8/5·10-s − 3.38·13-s − 16-s − 1.29·17-s − 0.0799·25-s + 3.38·26-s + 2.34·29-s + 1.29·34-s − 0.864·37-s + 8/5·40-s + 0.634·41-s + 1.75·49-s + 2/25·50-s + 1.96·53-s − 2.34·58-s − 0.524·61-s + 64-s − 5.41·65-s − 0.684·73-s + 0.864·74-s − 8/5·80-s − 0.634·82-s − 2.07·85-s − 0.0449·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8527427004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8527427004\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 86 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 95 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 479 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 470 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1175 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4406 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4079 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11714 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12578 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 43 T + p^{2} T^{2} )^{2} \) |
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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
−11.69492386544149924921055051079, −10.83587941109718151002143224722, −10.30459349199277242199574139107, −10.17957056069333076788603371001, −9.818292686284752372791545271000, −9.270903084822715937486195871929, −9.111161219969180311701950932927, −8.449316613956367250436138628535, −7.88501834338518742903421407481, −7.28877090078626510424250289342, −6.94112426043597471639473810578, −6.50002821114608599242113586901, −5.44839888069510333768065500214, −5.40092867674771557788398201201, −4.45137553475199564854127164155, −4.36513512652483728225204174722, −2.67475441402856615214680772067, −2.40898240298856797670141627345, −1.79725356768156053103726044034, −0.49719540401818754689173654570,
0.49719540401818754689173654570, 1.79725356768156053103726044034, 2.40898240298856797670141627345, 2.67475441402856615214680772067, 4.36513512652483728225204174722, 4.45137553475199564854127164155, 5.40092867674771557788398201201, 5.44839888069510333768065500214, 6.50002821114608599242113586901, 6.94112426043597471639473810578, 7.28877090078626510424250289342, 7.88501834338518742903421407481, 8.449316613956367250436138628535, 9.111161219969180311701950932927, 9.270903084822715937486195871929, 9.818292686284752372791545271000, 10.17957056069333076788603371001, 10.30459349199277242199574139107, 10.83587941109718151002143224722, 11.69492386544149924921055051079
