| L(s) = 1 | + 2-s + 4-s + 5-s − 8-s + 9-s + 10-s − 3·11-s + 8·13-s − 3·16-s + 18-s + 4·19-s + 20-s − 3·22-s + 2·23-s − 3·25-s + 8·26-s + 2·29-s − 5·32-s + 36-s + 4·38-s − 40-s + 10·41-s − 3·44-s + 45-s + 2·46-s + 4·47-s − 11·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 2.21·13-s − 3/4·16-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.639·22-s + 0.417·23-s − 3/5·25-s + 1.56·26-s + 0.371·29-s − 0.883·32-s + 1/6·36-s + 0.648·38-s − 0.158·40-s + 1.56·41-s − 0.452·44-s + 0.149·45-s + 0.294·46-s + 0.583·47-s − 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.561857817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.561857817\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - p T + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 181 T^{2} + p^{2} T^{4} \) |
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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
−9.694979283021973402117878407400, −9.231734017413645524959332326070, −8.545839841587401644388052383873, −8.344297489682947372193961689152, −7.49366161967159972181347201877, −7.15668445491862954026653685524, −6.35442098399677466546254811946, −5.94591452804038458123897994579, −5.70277406777468294406632128226, −4.92675699683738165625891258669, −4.31024380991304804281650407024, −3.54064825485925765036781384876, −3.09638401186146432093574745738, −2.24940545931227751655892724409, −1.22331264613907402948549628940,
1.22331264613907402948549628940, 2.24940545931227751655892724409, 3.09638401186146432093574745738, 3.54064825485925765036781384876, 4.31024380991304804281650407024, 4.92675699683738165625891258669, 5.70277406777468294406632128226, 5.94591452804038458123897994579, 6.35442098399677466546254811946, 7.15668445491862954026653685524, 7.49366161967159972181347201877, 8.344297489682947372193961689152, 8.545839841587401644388052383873, 9.231734017413645524959332326070, 9.694979283021973402117878407400
