| L(s) = 1 | − 2-s − 8·5-s − 7-s + 8-s + 8·10-s − 3·11-s − 5·13-s + 14-s − 16-s − 5·17-s − 3·19-s + 3·22-s + 6·23-s + 38·25-s + 5·26-s − 9·29-s − 8·31-s + 5·34-s + 8·35-s − 4·37-s + 3·38-s − 8·40-s + 5·41-s − 6·46-s + 6·47-s − 38·50-s + 22·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3.57·5-s − 0.377·7-s + 0.353·8-s + 2.52·10-s − 0.904·11-s − 1.38·13-s + 0.267·14-s − 1/4·16-s − 1.21·17-s − 0.688·19-s + 0.639·22-s + 1.25·23-s + 38/5·25-s + 0.980·26-s − 1.67·29-s − 1.43·31-s + 0.857·34-s + 1.35·35-s − 0.657·37-s + 0.486·38-s − 1.26·40-s + 0.780·41-s − 0.884·46-s + 0.875·47-s − 5.37·50-s + 3.02·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2950970532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2950970532\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.209572250965169719824095895726, −9.076285994286354126864852649807, −8.730989720272748181507728750806, −8.455029091833649262596810241522, −7.83850930514762571768700933667, −7.58075107221998171421476051498, −7.35076259395235782952250669161, −7.16388363282129972261873711635, −6.76724997137839378946880176352, −6.04490230441056696329922628557, −5.06776879853658935663386643013, −5.02721514047159016780863977095, −4.61205922556751389052858201972, −3.87917850940955487133888764526, −3.71265008499387185484580174349, −3.47043946501003338384843077762, −2.43221517391596455348626634829, −2.29583356565372362450505337540, −0.53510965186498306838775895806, −0.50096710904694089067719525983,
0.50096710904694089067719525983, 0.53510965186498306838775895806, 2.29583356565372362450505337540, 2.43221517391596455348626634829, 3.47043946501003338384843077762, 3.71265008499387185484580174349, 3.87917850940955487133888764526, 4.61205922556751389052858201972, 5.02721514047159016780863977095, 5.06776879853658935663386643013, 6.04490230441056696329922628557, 6.76724997137839378946880176352, 7.16388363282129972261873711635, 7.35076259395235782952250669161, 7.58075107221998171421476051498, 7.83850930514762571768700933667, 8.455029091833649262596810241522, 8.730989720272748181507728750806, 9.076285994286354126864852649807, 9.209572250965169719824095895726
