L(s) = 1 | + 2-s − 3-s − 2·5-s − 6-s − 8-s − 2·10-s − 11-s + 2·13-s + 2·15-s − 16-s + 2·17-s − 22-s − 23-s + 24-s + 3·25-s + 2·26-s + 27-s − 29-s + 2·30-s − 2·31-s + 33-s + 2·34-s + 37-s − 2·39-s + 2·40-s + 43-s − 46-s + ⋯ |
L(s) = 1 | + 2-s − 3-s − 2·5-s − 6-s − 8-s − 2·10-s − 11-s + 2·13-s + 2·15-s − 16-s + 2·17-s − 22-s − 23-s + 24-s + 3·25-s + 2·26-s + 27-s − 29-s + 2·30-s − 2·31-s + 33-s + 2·34-s + 37-s − 2·39-s + 2·40-s + 43-s − 46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5714698676\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5714698676\) |
\(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.974784129365682638359242953395, −9.168553505774915029795332745985, −9.123490154657383279849162407230, −8.352971208711085777644249772539, −8.344047369398747222890570315653, −7.72631677512436013508027944563, −7.36272270622419921711122879124, −7.22218939015665074314621385008, −6.25395447626052788118103128782, −6.02318342050676670170937782801, −5.59166644393434739119604911868, −5.52104663999998049994575977915, −4.75654329776446243524431860871, −4.40237697517048307052104126073, −3.86577231823986330470112004461, −3.63596889452085221097128607364, −3.16942435994920246102566525812, −2.74892321565279398541930181665, −1.47940075909707004064507090842, −0.58333095216195354247331909647,
0.58333095216195354247331909647, 1.47940075909707004064507090842, 2.74892321565279398541930181665, 3.16942435994920246102566525812, 3.63596889452085221097128607364, 3.86577231823986330470112004461, 4.40237697517048307052104126073, 4.75654329776446243524431860871, 5.52104663999998049994575977915, 5.59166644393434739119604911868, 6.02318342050676670170937782801, 6.25395447626052788118103128782, 7.22218939015665074314621385008, 7.36272270622419921711122879124, 7.72631677512436013508027944563, 8.344047369398747222890570315653, 8.352971208711085777644249772539, 9.123490154657383279849162407230, 9.168553505774915029795332745985, 9.974784129365682638359242953395