| L(s) = 1 | − 2-s − 3·4-s + 7·8-s + 5·16-s − 28·17-s − 33·32-s + 28·34-s + 98·49-s − 172·53-s + 236·61-s + 13·64-s + 84·68-s − 98·98-s + 172·106-s + 44·109-s + 412·113-s + 242·121-s − 236·122-s + 127-s + 119·128-s + 131-s − 196·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·2-s − 3/4·4-s + 7/8·8-s + 5/16·16-s − 1.64·17-s − 1.03·32-s + 0.823·34-s + 2·49-s − 3.24·53-s + 3.86·61-s + 0.203·64-s + 1.23·68-s − 98-s + 1.62·106-s + 0.403·109-s + 3.64·113-s + 2·121-s − 1.93·122-s + 0.00787·127-s + 0.929·128-s + 0.00763·131-s − 1.44·136-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9122505128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9122505128\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 118 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 98 T + p^{2} T^{2} )( 1 + 98 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 154 T + p^{2} T^{2} )( 1 + 154 T + p^{2} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
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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
−9.982247680809211275624759560386, −9.796638187131818344652065904701, −9.201177531054721278507405034194, −8.777892543664008374735245398425, −8.632754257181835064400692993148, −8.176042760982888632974161853258, −7.57161792660906148900417178896, −7.31333340901069322366069550023, −6.69538806193598826037447788985, −6.38084833664336189937780467145, −5.72306951877626974956730406173, −5.29156776243778550027200244436, −4.60119427058493335107891428006, −4.52803036804800300980522416622, −3.77188145985752248921131548711, −3.39437501238140928989555702107, −2.43324538594531939075421903858, −2.03839226250484667549601661585, −1.13844882646762746321204986121, −0.38719970428042186630234326727,
0.38719970428042186630234326727, 1.13844882646762746321204986121, 2.03839226250484667549601661585, 2.43324538594531939075421903858, 3.39437501238140928989555702107, 3.77188145985752248921131548711, 4.52803036804800300980522416622, 4.60119427058493335107891428006, 5.29156776243778550027200244436, 5.72306951877626974956730406173, 6.38084833664336189937780467145, 6.69538806193598826037447788985, 7.31333340901069322366069550023, 7.57161792660906148900417178896, 8.176042760982888632974161853258, 8.632754257181835064400692993148, 8.777892543664008374735245398425, 9.201177531054721278507405034194, 9.796638187131818344652065904701, 9.982247680809211275624759560386
