| L(s) = 1 | − 2·7-s + 6·9-s − 4·17-s − 8·23-s + 10·25-s + 16·31-s + 4·41-s + 3·49-s − 12·63-s + 28·73-s + 8·79-s + 27·81-s + 12·89-s + 12·97-s − 16·103-s + 12·113-s + 8·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 16·161-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2·9-s − 0.970·17-s − 1.66·23-s + 2·25-s + 2.87·31-s + 0.624·41-s + 3/7·49-s − 1.51·63-s + 3.27·73-s + 0.900·79-s + 3·81-s + 1.27·89-s + 1.21·97-s − 1.57·103-s + 1.12·113-s + 0.733·119-s + 1.63·121-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 − 1.94·153-s + 0.0798·157-s + 1.26·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.775972660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.775972660\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.393367676538779167496785765384, −9.320209239220672871166500492954, −8.689966385491247594680954675670, −8.271106662033107465754436522294, −7.80355083781785591803402612706, −7.64273674400751245335509083138, −6.81478805656256983583772616805, −6.75269690709711178518935595228, −6.42676866958354641830924857074, −6.14240338586608804719016598365, −5.29046311001902558318243798540, −4.90409942133053111483083596142, −4.34717606299989114365403881882, −4.30869328972446403094224069869, −3.62428761066830341936472806156, −3.16967327530384778830534848535, −2.35312608472753416730579360560, −2.17799861782364476694804558037, −1.18092260019511124532738778402, −0.71815569499597098937551368353,
0.71815569499597098937551368353, 1.18092260019511124532738778402, 2.17799861782364476694804558037, 2.35312608472753416730579360560, 3.16967327530384778830534848535, 3.62428761066830341936472806156, 4.30869328972446403094224069869, 4.34717606299989114365403881882, 4.90409942133053111483083596142, 5.29046311001902558318243798540, 6.14240338586608804719016598365, 6.42676866958354641830924857074, 6.75269690709711178518935595228, 6.81478805656256983583772616805, 7.64273674400751245335509083138, 7.80355083781785591803402612706, 8.271106662033107465754436522294, 8.689966385491247594680954675670, 9.320209239220672871166500492954, 9.393367676538779167496785765384
