L(s) = 1 | + 3-s + 9-s − 4·13-s + 8·23-s + 25-s + 27-s + 12·37-s − 4·39-s − 8·47-s + 2·49-s + 16·59-s − 4·61-s + 8·69-s + 16·71-s − 12·73-s + 75-s + 81-s + 8·83-s + 4·97-s − 8·107-s − 4·109-s + 12·111-s − 4·117-s − 6·121-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.10·13-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.97·37-s − 0.640·39-s − 1.16·47-s + 2/7·49-s + 2.08·59-s − 0.512·61-s + 0.963·69-s + 1.89·71-s − 1.40·73-s + 0.115·75-s + 1/9·81-s + 0.878·83-s + 0.406·97-s − 0.773·107-s − 0.383·109-s + 1.13·111-s − 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.000768501\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.000768501\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.291298574402970679863122371973, −8.621666342891523369321526585402, −8.360951480912939394668117342589, −7.59137484122417926942388693646, −7.46466097862180301658442852635, −6.74701028543655247264600767887, −6.46317561196916208708331018309, −5.60216332901476637074044973637, −5.10776393049599874582799242162, −4.61517774336118835532603327204, −4.03845309884361087663949621816, −3.23381133409479097641702786283, −2.72724935346905167341287416802, −2.08564612966020829296841400230, −0.932124893814943855253891093379,
0.932124893814943855253891093379, 2.08564612966020829296841400230, 2.72724935346905167341287416802, 3.23381133409479097641702786283, 4.03845309884361087663949621816, 4.61517774336118835532603327204, 5.10776393049599874582799242162, 5.60216332901476637074044973637, 6.46317561196916208708331018309, 6.74701028543655247264600767887, 7.46466097862180301658442852635, 7.59137484122417926942388693646, 8.360951480912939394668117342589, 8.621666342891523369321526585402, 9.291298574402970679863122371973