| L(s) = 1 | + 2·3-s + 6·5-s + 3·9-s − 6·11-s + 12·15-s + 2·19-s − 6·23-s + 19·25-s + 10·27-s + 12·29-s − 8·31-s − 12·33-s + 2·37-s + 18·45-s − 6·53-s − 36·55-s + 4·57-s + 6·59-s − 6·61-s − 6·67-s − 12·69-s + 12·73-s + 38·75-s + 6·79-s + 20·81-s + 12·83-s + 24·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.68·5-s + 9-s − 1.80·11-s + 3.09·15-s + 0.458·19-s − 1.25·23-s + 19/5·25-s + 1.92·27-s + 2.22·29-s − 1.43·31-s − 2.08·33-s + 0.328·37-s + 2.68·45-s − 0.824·53-s − 4.85·55-s + 0.529·57-s + 0.781·59-s − 0.768·61-s − 0.733·67-s − 1.44·69-s + 1.40·73-s + 4.38·75-s + 0.675·79-s + 20/9·81-s + 1.31·83-s + 2.57·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.963694883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.963694883\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−10.35790841082252018411878541089, −10.11040732871490540876658443469, −9.590002472144097237106077941810, −9.275721755863283411188510091002, −9.059069206134996321125605381393, −8.295879663483255015463289808722, −8.072087805167689252802864249116, −7.70373802667479973760245446603, −6.93615822123185691673378864227, −6.54334168776319052839973059571, −6.13461265635285938543077954205, −5.63886776221557643105285435924, −5.05804684329085506654964567027, −4.97173540299177572649287045373, −4.12101457259943252118732261643, −3.15159152585006383867867430106, −2.86608744029093119602358986889, −2.16294737197199234163774045176, −2.08224269043742124058466608210, −1.11927073919964896646083367438,
1.11927073919964896646083367438, 2.08224269043742124058466608210, 2.16294737197199234163774045176, 2.86608744029093119602358986889, 3.15159152585006383867867430106, 4.12101457259943252118732261643, 4.97173540299177572649287045373, 5.05804684329085506654964567027, 5.63886776221557643105285435924, 6.13461265635285938543077954205, 6.54334168776319052839973059571, 6.93615822123185691673378864227, 7.70373802667479973760245446603, 8.072087805167689252802864249116, 8.295879663483255015463289808722, 9.059069206134996321125605381393, 9.275721755863283411188510091002, 9.590002472144097237106077941810, 10.11040732871490540876658443469, 10.35790841082252018411878541089
