| L(s) = 1 | + 2·4-s − 4·9-s + 20·11-s + 3·16-s − 8·19-s + 12·29-s − 12·31-s − 8·36-s + 40·44-s + 22·49-s − 36·59-s + 4·61-s + 12·64-s + 8·71-s − 16·76-s + 24·79-s − 6·81-s − 80·99-s − 4·101-s + 32·109-s + 24·116-s + 210·121-s − 24·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4-s − 4/3·9-s + 6.03·11-s + 3/4·16-s − 1.83·19-s + 2.22·29-s − 2.15·31-s − 4/3·36-s + 6.03·44-s + 22/7·49-s − 4.68·59-s + 0.512·61-s + 3/2·64-s + 0.949·71-s − 1.83·76-s + 2.70·79-s − 2/3·81-s − 8.04·99-s − 0.398·101-s + 3.06·109-s + 2.22·116-s + 19.0·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.817818677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.817818677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 10 T + 45 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1171 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 9046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 8707 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 18 T + 181 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 9939 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 118 T^{2} + 13731 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 364 T^{2} + 51814 T^{4} - 364 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.459930384049643570281722043643, −8.325860495597171295581448654450, −8.166719132099062110869062307269, −7.39461066895235777187870420412, −7.34109316110258255757843631137, −7.08124667981461412770921691061, −6.85635021326030571151531905124, −6.55452292942833195223493985951, −6.29317014080714225484491910677, −6.21165613770595484630639246183, −6.06730956451145665669966274307, −5.87704480969394974254302155295, −5.39998033436526572843556361016, −4.77784714442873006714561120787, −4.54950158803876015371852782887, −4.36565834943948178421770626874, −3.86284377154361413317387756338, −3.75626631839099463464771980300, −3.44948139045478085637840925320, −3.30877798068781622526431988584, −2.45730092961093258552906110465, −2.28125934635601066399765182316, −1.75329039308780820149713557272, −1.29601766580409398498140242963, −1.01635714206667413712278491866,
1.01635714206667413712278491866, 1.29601766580409398498140242963, 1.75329039308780820149713557272, 2.28125934635601066399765182316, 2.45730092961093258552906110465, 3.30877798068781622526431988584, 3.44948139045478085637840925320, 3.75626631839099463464771980300, 3.86284377154361413317387756338, 4.36565834943948178421770626874, 4.54950158803876015371852782887, 4.77784714442873006714561120787, 5.39998033436526572843556361016, 5.87704480969394974254302155295, 6.06730956451145665669966274307, 6.21165613770595484630639246183, 6.29317014080714225484491910677, 6.55452292942833195223493985951, 6.85635021326030571151531905124, 7.08124667981461412770921691061, 7.34109316110258255757843631137, 7.39461066895235777187870420412, 8.166719132099062110869062307269, 8.325860495597171295581448654450, 8.459930384049643570281722043643
