L(s) = 1 | − 2-s + 5-s − 7-s + 8-s − 10-s + 14-s − 16-s + 23-s − 29-s − 35-s + 40-s − 41-s + 2·43-s − 46-s + 47-s + 49-s − 56-s + 58-s + 61-s + 64-s − 67-s + 70-s − 80-s + 82-s + 83-s − 2·86-s + 2·89-s + ⋯ |
L(s) = 1 | − 2-s + 5-s − 7-s + 8-s − 10-s + 14-s − 16-s + 23-s − 29-s − 35-s + 40-s − 41-s + 2·43-s − 46-s + 47-s + 49-s − 56-s + 58-s + 61-s + 64-s − 67-s + 70-s − 80-s + 82-s + 83-s − 2·86-s + 2·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4608339515\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4608339515\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
good | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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
−10.97017062096837622727523026908, −10.62593273313949135147153047916, −10.27620481528115088537215819704, −9.872053017151147102207854555176, −9.329979014909354021152153654827, −9.232752757954356762436883107694, −8.832143153405060182665829442173, −8.319425501913797546385383074319, −7.52519517997436693382247180723, −7.41238559431710544597151247096, −6.83451333307522758265591328945, −6.20544926634159153008711595560, −5.89319377979588682701216326736, −5.24649080765084844197287196045, −4.79864218464574690665485036118, −3.96557977926722032178687934316, −3.52664835806404417048596800116, −2.59372777403914185558863819939, −2.07320807731151890144516204983, −1.06818385127464349816642610461,
1.06818385127464349816642610461, 2.07320807731151890144516204983, 2.59372777403914185558863819939, 3.52664835806404417048596800116, 3.96557977926722032178687934316, 4.79864218464574690665485036118, 5.24649080765084844197287196045, 5.89319377979588682701216326736, 6.20544926634159153008711595560, 6.83451333307522758265591328945, 7.41238559431710544597151247096, 7.52519517997436693382247180723, 8.319425501913797546385383074319, 8.832143153405060182665829442173, 9.232752757954356762436883107694, 9.329979014909354021152153654827, 9.872053017151147102207854555176, 10.27620481528115088537215819704, 10.62593273313949135147153047916, 10.97017062096837622727523026908