| L(s) = 1 | − 3-s − 4-s + 2·5-s − 2·11-s + 12-s + 13-s − 2·15-s + 17-s − 2·20-s + 3·25-s + 27-s − 2·29-s + 2·33-s − 39-s + 2·44-s + 47-s − 51-s − 52-s − 4·55-s + 2·60-s + 64-s + 2·65-s − 68-s − 2·71-s + 73-s − 3·75-s + 79-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 2·5-s − 2·11-s + 12-s + 13-s − 2·15-s + 17-s − 2·20-s + 3·25-s + 27-s − 2·29-s + 2·33-s − 39-s + 2·44-s + 47-s − 51-s − 52-s − 4·55-s + 2·60-s + 64-s + 2·65-s − 68-s − 2·71-s + 73-s − 3·75-s + 79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8131146299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8131146299\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.391543915944508440803028473134, −9.113039009713901438689183647457, −8.651479263543325687484549024903, −8.598357303782226854378435486374, −7.72863264483160389278303620991, −7.63336708363880330355617761327, −7.09752147932774029851318586488, −6.36880838177893281941886311643, −6.21892941143911630894719812732, −5.71384383811787228697317398951, −5.47656027123747436087920765821, −5.31847691003586788130934740913, −4.74326881169303380057128796801, −4.57037316011366837436553061345, −3.49500381135021437934779937940, −3.40261795043505058658331050062, −2.46467620292673372629862932994, −2.26883091674488514851389928509, −1.47249227984080015289716550848, −0.70294758872937407203185684283,
0.70294758872937407203185684283, 1.47249227984080015289716550848, 2.26883091674488514851389928509, 2.46467620292673372629862932994, 3.40261795043505058658331050062, 3.49500381135021437934779937940, 4.57037316011366837436553061345, 4.74326881169303380057128796801, 5.31847691003586788130934740913, 5.47656027123747436087920765821, 5.71384383811787228697317398951, 6.21892941143911630894719812732, 6.36880838177893281941886311643, 7.09752147932774029851318586488, 7.63336708363880330355617761327, 7.72863264483160389278303620991, 8.598357303782226854378435486374, 8.651479263543325687484549024903, 9.113039009713901438689183647457, 9.391543915944508440803028473134
