L(s) = 1 | − 3-s + 9-s + 11-s + 4·13-s + 2·25-s − 27-s − 33-s + 4·37-s − 4·39-s + 12·47-s + 2·49-s + 12·59-s − 8·61-s − 12·71-s − 8·73-s − 2·75-s + 81-s − 20·97-s + 99-s + 4·109-s − 4·111-s + 4·117-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 2/5·25-s − 0.192·27-s − 0.174·33-s + 0.657·37-s − 0.640·39-s + 1.75·47-s + 2/7·49-s + 1.56·59-s − 1.02·61-s − 1.42·71-s − 0.936·73-s − 0.230·75-s + 1/9·81-s − 2.03·97-s + 0.100·99-s + 0.383·109-s − 0.379·111-s + 0.369·117-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.008812861\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.008812861\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.87111583454372921305705022754, −10.57755026186379514837066820890, −9.872822078123312712017635366132, −9.339855126610289039276263015751, −8.670410637699767160646012915426, −8.342849935134971891357488978271, −7.38133321672302546420095787319, −7.07489979355103692195249240932, −6.13113280286443542004063358489, −5.94054070785065555014940896467, −5.09827818531269205265502168789, −4.29213337932568858922592669887, −3.70936690680474055080283033333, −2.63227716153411648714898260305, −1.25821432380895100299434540193,
1.25821432380895100299434540193, 2.63227716153411648714898260305, 3.70936690680474055080283033333, 4.29213337932568858922592669887, 5.09827818531269205265502168789, 5.94054070785065555014940896467, 6.13113280286443542004063358489, 7.07489979355103692195249240932, 7.38133321672302546420095787319, 8.342849935134971891357488978271, 8.670410637699767160646012915426, 9.339855126610289039276263015751, 9.872822078123312712017635366132, 10.57755026186379514837066820890, 10.87111583454372921305705022754