| L(s) = 1 | − 4-s + 2·5-s + 7-s − 11-s + 2·17-s + 2·19-s − 2·20-s − 23-s + 25-s − 28-s + 2·35-s + 43-s + 44-s − 47-s + 49-s − 2·55-s + 61-s + 64-s − 2·68-s − 2·73-s − 2·76-s − 77-s + 2·83-s + 4·85-s + 92-s + 4·95-s − 100-s + ⋯ |
| L(s) = 1 | − 4-s + 2·5-s + 7-s − 11-s + 2·17-s + 2·19-s − 2·20-s − 23-s + 25-s − 28-s + 2·35-s + 43-s + 44-s − 47-s + 49-s − 2·55-s + 61-s + 64-s − 2·68-s − 2·73-s − 2·76-s − 77-s + 2·83-s + 4·85-s + 92-s + 4·95-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.511495860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511495860\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.955146669020100099234226323524, −9.473350366130719601899813161456, −9.267016579407686003073661140496, −8.688641296225293314978865255507, −8.282249022368222087371795885205, −7.82044468243207811765996315903, −7.57377715365348270152309666477, −7.25223459027173445593860181239, −6.41565335976116401627386509961, −5.91562007881711043693400735385, −5.66063028445808983305019119877, −5.40007072398068789325028412127, −4.85292776070008308252552085477, −4.84111666168071033696443910353, −3.68863392047217206878572793539, −3.66804513659111975324488354019, −2.53988542889064469549143382216, −2.48586216376119661316447639978, −1.51032557721313214793448982898, −1.19677184367361732318077074642,
1.19677184367361732318077074642, 1.51032557721313214793448982898, 2.48586216376119661316447639978, 2.53988542889064469549143382216, 3.66804513659111975324488354019, 3.68863392047217206878572793539, 4.84111666168071033696443910353, 4.85292776070008308252552085477, 5.40007072398068789325028412127, 5.66063028445808983305019119877, 5.91562007881711043693400735385, 6.41565335976116401627386509961, 7.25223459027173445593860181239, 7.57377715365348270152309666477, 7.82044468243207811765996315903, 8.282249022368222087371795885205, 8.688641296225293314978865255507, 9.267016579407686003073661140496, 9.473350366130719601899813161456, 9.955146669020100099234226323524
