| L(s) = 1 | − 10·13-s + 10·17-s + 10·37-s + 16·41-s + 10·53-s + 24·61-s − 10·73-s − 9·81-s − 10·97-s − 4·101-s − 30·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2.77·13-s + 2.42·17-s + 1.64·37-s + 2.49·41-s + 1.37·53-s + 3.07·61-s − 1.17·73-s − 81-s − 1.01·97-s − 0.398·101-s − 2.82·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.056160146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056160146\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−9.601592281465952282014532857786, −9.500237568668142842893983551188, −8.851430931675751862171392110095, −8.287930654625881369253654658357, −7.918244794743921448562385195935, −7.61040917803321066918598047893, −7.21314894823549302892280489246, −7.07564203753473218927458952273, −6.38341562420334819823389927008, −5.72123793238406465915987156058, −5.49081465104499543685072245073, −5.28335749916440827837260673826, −4.42967311077520984884769525126, −4.37204691372230035009015211978, −3.66124572424794132344560121879, −3.01561917309734725438421389998, −2.50266323062434627352120824100, −2.33298662812018307122173189170, −1.22659962426848624835611065649, −0.61260532860234294807589608905,
0.61260532860234294807589608905, 1.22659962426848624835611065649, 2.33298662812018307122173189170, 2.50266323062434627352120824100, 3.01561917309734725438421389998, 3.66124572424794132344560121879, 4.37204691372230035009015211978, 4.42967311077520984884769525126, 5.28335749916440827837260673826, 5.49081465104499543685072245073, 5.72123793238406465915987156058, 6.38341562420334819823389927008, 7.07564203753473218927458952273, 7.21314894823549302892280489246, 7.61040917803321066918598047893, 7.918244794743921448562385195935, 8.287930654625881369253654658357, 8.851430931675751862171392110095, 9.500237568668142842893983551188, 9.601592281465952282014532857786
