| L(s) = 1 | + 2-s + 4-s + 8-s − 2·13-s + 16-s − 25-s − 2·26-s + 32-s − 2·37-s + 24·47-s + 2·49-s − 50-s − 2·52-s − 2·61-s + 64-s + 24·71-s + 22·73-s − 2·74-s + 24·83-s + 24·94-s + 4·97-s + 2·98-s − 100-s − 2·104-s − 24·107-s + 22·109-s − 22·121-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.554·13-s + 1/4·16-s − 1/5·25-s − 0.392·26-s + 0.176·32-s − 0.328·37-s + 3.50·47-s + 2/7·49-s − 0.141·50-s − 0.277·52-s − 0.256·61-s + 1/8·64-s + 2.84·71-s + 2.57·73-s − 0.232·74-s + 2.63·83-s + 2.47·94-s + 0.406·97-s + 0.202·98-s − 0.0999·100-s − 0.196·104-s − 2.32·107-s + 2.10·109-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.612283659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.612283659\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−9.101601246278505892226541271616, −8.542414927665885183690210552564, −8.047221009677602422962699873621, −7.44571478799237129140639120122, −7.24345462277249174674167705434, −6.50974172394367970294419909175, −6.18735538198950166736525445007, −5.49090131952205180218546644094, −5.10609983558079824652464594116, −4.60529471992634973001671480527, −3.69589098584017718744945020548, −3.68824329939728705086964001391, −2.41615112845105918353887046447, −2.30016225881827778465495235815, −0.935395355311356120178486862505,
0.935395355311356120178486862505, 2.30016225881827778465495235815, 2.41615112845105918353887046447, 3.68824329939728705086964001391, 3.69589098584017718744945020548, 4.60529471992634973001671480527, 5.10609983558079824652464594116, 5.49090131952205180218546644094, 6.18735538198950166736525445007, 6.50974172394367970294419909175, 7.24345462277249174674167705434, 7.44571478799237129140639120122, 8.047221009677602422962699873621, 8.542414927665885183690210552564, 9.101601246278505892226541271616
