L(s) = 1 | + 8·7-s + 9-s − 12·17-s − 6·25-s − 8·31-s + 4·41-s − 16·47-s + 34·49-s + 8·63-s + 32·71-s − 12·73-s − 8·79-s + 81-s + 20·89-s − 28·97-s + 24·103-s + 4·113-s − 96·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + ⋯ |
L(s) = 1 | + 3.02·7-s + 1/3·9-s − 2.91·17-s − 6/5·25-s − 1.43·31-s + 0.624·41-s − 2.33·47-s + 34/7·49-s + 1.00·63-s + 3.79·71-s − 1.40·73-s − 0.900·79-s + 1/9·81-s + 2.11·89-s − 2.84·97-s + 2.36·103-s + 0.376·113-s − 8.80·119-s − 0.545·121-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 − 0.970·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.417262496\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.417262496\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−11.01151255843213055292894337270, −10.84849442743057013529606410604, −9.826637660192491869357018654726, −9.240123327709471504547415939693, −8.460175078145674768211752918036, −8.391046533691483698004382219288, −7.67780945046588747720337476950, −7.17212008872630953090710242182, −6.44896944953500655110243120870, −5.56983356275681900785379754785, −4.77602747706575131165960838209, −4.62570703678849832502580328887, −3.85566973953101374011358603281, −2.09803110754033007053716721218, −1.83689405023894510739841616557,
1.83689405023894510739841616557, 2.09803110754033007053716721218, 3.85566973953101374011358603281, 4.62570703678849832502580328887, 4.77602747706575131165960838209, 5.56983356275681900785379754785, 6.44896944953500655110243120870, 7.17212008872630953090710242182, 7.67780945046588747720337476950, 8.391046533691483698004382219288, 8.460175078145674768211752918036, 9.240123327709471504547415939693, 9.826637660192491869357018654726, 10.84849442743057013529606410604, 11.01151255843213055292894337270