| L(s) = 1 | + 4·7-s − 9-s + 4·17-s − 25-s + 2·31-s + 2·41-s + 10·49-s − 4·63-s − 2·73-s + 2·97-s + 16·119-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s − 4·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 4·7-s − 9-s + 4·17-s − 25-s + 2·31-s + 2·41-s + 10·49-s − 4·63-s − 2·73-s + 2·97-s + 16·119-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s − 4·175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.641926122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.641926122\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.93513208391249524060185296544, −5.85186660711243372905135614546, −5.80113768708420282360123574107, −5.57893420695897229207622306939, −5.44680303014830919589535263719, −5.18469459265082166997242566710, −5.03216854716428668642060173521, −4.84430981138075124449622775563, −4.60164786023931492249245285627, −4.52203560917423071567810775307, −4.27887544337917475954071656420, −4.02326968539852381565238900161, −3.64511799659171553158835914152, −3.57189688292854495768567703790, −3.55185959981753435916359257989, −2.88624268658424691385516941650, −2.69590975403555762289568677486, −2.68638259072815263849897155521, −2.42015973939543157762047423402, −2.01237503475484045843156791844, −1.80917207467162596516210792736, −1.44197080686420177861433626896, −1.16836248226138173994248206264, −1.01955770609237974815582454792, −0.984455087445693683622800858141,
0.984455087445693683622800858141, 1.01955770609237974815582454792, 1.16836248226138173994248206264, 1.44197080686420177861433626896, 1.80917207467162596516210792736, 2.01237503475484045843156791844, 2.42015973939543157762047423402, 2.68638259072815263849897155521, 2.69590975403555762289568677486, 2.88624268658424691385516941650, 3.55185959981753435916359257989, 3.57189688292854495768567703790, 3.64511799659171553158835914152, 4.02326968539852381565238900161, 4.27887544337917475954071656420, 4.52203560917423071567810775307, 4.60164786023931492249245285627, 4.84430981138075124449622775563, 5.03216854716428668642060173521, 5.18469459265082166997242566710, 5.44680303014830919589535263719, 5.57893420695897229207622306939, 5.80113768708420282360123574107, 5.85186660711243372905135614546, 5.93513208391249524060185296544
