| L(s) = 1 | − 2-s + 3-s − 6-s − 5·7-s + 8-s − 3·11-s + 8·13-s + 5·14-s − 16-s + 4·19-s − 5·21-s + 3·22-s + 24-s − 8·26-s − 27-s + 18·29-s + 31-s − 3·33-s + 8·37-s − 4·38-s + 8·39-s + 5·42-s + 20·43-s − 6·47-s − 48-s + 18·49-s − 3·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s − 1.88·7-s + 0.353·8-s − 0.904·11-s + 2.21·13-s + 1.33·14-s − 1/4·16-s + 0.917·19-s − 1.09·21-s + 0.639·22-s + 0.204·24-s − 1.56·26-s − 0.192·27-s + 3.34·29-s + 0.179·31-s − 0.522·33-s + 1.31·37-s − 0.648·38-s + 1.28·39-s + 0.771·42-s + 3.04·43-s − 0.875·47-s − 0.144·48-s + 18/7·49-s − 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536986951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536986951\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−9.973563051449953779557160817183, −9.744858627252230756232203674968, −9.101975859263661736758333345918, −9.062954095047115403321517442642, −8.376748417254120533624820626034, −8.364684890514208047091986603515, −7.54134283671819224331255027361, −7.52704049057369605583385235807, −6.57390257645950109053749176522, −6.49090782991453514098084755022, −5.88990624408301048799004922197, −5.75848231558810287171853896033, −4.77574269630849804260675490790, −4.37614593430037840563207157328, −3.69435296552545415382590779630, −3.29475552576276514913268018561, −2.78014600706319245005821475496, −2.46934182278326863449203643206, −1.13457140669304745613012047354, −0.74442779376075173490285700792,
0.74442779376075173490285700792, 1.13457140669304745613012047354, 2.46934182278326863449203643206, 2.78014600706319245005821475496, 3.29475552576276514913268018561, 3.69435296552545415382590779630, 4.37614593430037840563207157328, 4.77574269630849804260675490790, 5.75848231558810287171853896033, 5.88990624408301048799004922197, 6.49090782991453514098084755022, 6.57390257645950109053749176522, 7.52704049057369605583385235807, 7.54134283671819224331255027361, 8.364684890514208047091986603515, 8.376748417254120533624820626034, 9.062954095047115403321517442642, 9.101975859263661736758333345918, 9.744858627252230756232203674968, 9.973563051449953779557160817183
