| L(s) = 1 | + 3·9-s − 16·11-s − 16·19-s + 26·29-s + 14·31-s − 6·41-s − 8·49-s − 8·59-s + 28·61-s − 10·71-s + 8·79-s − 7·81-s − 40·89-s − 48·99-s − 32·101-s + 116·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 39·169-s − 48·171-s + ⋯ |
| L(s) = 1 | + 9-s − 4.82·11-s − 3.67·19-s + 4.82·29-s + 2.51·31-s − 0.937·41-s − 8/7·49-s − 1.04·59-s + 3.58·61-s − 1.18·71-s + 0.900·79-s − 7/9·81-s − 4.23·89-s − 4.82·99-s − 3.18·101-s + 10.5·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3·169-s − 3.67·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5435977056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5435977056\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 46 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 680 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 766 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4430 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3934 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 99 T^{2} + 5912 T^{4} - 99 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1750 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11734 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 137 T^{2} + 11776 T^{4} + 137 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 5254 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.85967092972510619297879293103, −5.68833009965830259137738192201, −5.37246112457987608088043507193, −5.26975898131392061225082857582, −4.95863565939844746628080249133, −4.81190604587418787663022566939, −4.75650403919347359224137648261, −4.58019422884141578344468285036, −4.46556837653542659029556557744, −4.17034983856584473864492827232, −3.99905550864693684333654420467, −3.58350835055198882615681783056, −3.57741243969026835730123204641, −2.89187067048410169559233977724, −2.84627001412334676228937280838, −2.62283112932655061395816355007, −2.52392607575170497083776337443, −2.49784827683864749348359778673, −2.47411355451289419134388882183, −1.81256664893155474710845073819, −1.50736981293879733787855733832, −1.29158433628777056655537454814, −0.928492083624753121380671307462, −0.38589458470359794684406099841, −0.16962741683272865265108465500,
0.16962741683272865265108465500, 0.38589458470359794684406099841, 0.928492083624753121380671307462, 1.29158433628777056655537454814, 1.50736981293879733787855733832, 1.81256664893155474710845073819, 2.47411355451289419134388882183, 2.49784827683864749348359778673, 2.52392607575170497083776337443, 2.62283112932655061395816355007, 2.84627001412334676228937280838, 2.89187067048410169559233977724, 3.57741243969026835730123204641, 3.58350835055198882615681783056, 3.99905550864693684333654420467, 4.17034983856584473864492827232, 4.46556837653542659029556557744, 4.58019422884141578344468285036, 4.75650403919347359224137648261, 4.81190604587418787663022566939, 4.95863565939844746628080249133, 5.26975898131392061225082857582, 5.37246112457987608088043507193, 5.68833009965830259137738192201, 5.85967092972510619297879293103
