| L(s) = 1 | + 2·3-s − 3·9-s + 6·11-s + 16·17-s − 9·25-s − 14·27-s + 12·33-s + 4·41-s + 22·43-s − 10·49-s + 32·51-s + 24·67-s + 8·73-s − 18·75-s − 4·81-s − 8·83-s − 20·89-s − 4·97-s − 18·99-s + 4·107-s − 12·113-s + 5·121-s + 8·123-s + 127-s + 44·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 9-s + 1.80·11-s + 3.88·17-s − 9/5·25-s − 2.69·27-s + 2.08·33-s + 0.624·41-s + 3.35·43-s − 1.42·49-s + 4.48·51-s + 2.93·67-s + 0.936·73-s − 2.07·75-s − 4/9·81-s − 0.878·83-s − 2.11·89-s − 0.406·97-s − 1.80·99-s + 0.386·107-s − 1.12·113-s + 5/11·121-s + 0.721·123-s + 0.0887·127-s + 3.87·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.531361756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.531361756\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.146037049670989337122206299194, −7.75842963304955327229332692564, −7.68992355213504990402216682341, −7.02533007960674119090827223190, −6.24198317410599431388761026596, −5.97874250935484417499511635389, −5.45096814093615086314212064193, −5.35705266569945906014064710560, −4.12034263164171481182071378887, −3.80259981482385994331528030455, −3.53944000086915839540994955829, −2.91973338015050902044564271809, −2.44287708620897533980700790337, −1.55945640908010126049243572055, −0.907460307275556682567938804358,
0.907460307275556682567938804358, 1.55945640908010126049243572055, 2.44287708620897533980700790337, 2.91973338015050902044564271809, 3.53944000086915839540994955829, 3.80259981482385994331528030455, 4.12034263164171481182071378887, 5.35705266569945906014064710560, 5.45096814093615086314212064193, 5.97874250935484417499511635389, 6.24198317410599431388761026596, 7.02533007960674119090827223190, 7.68992355213504990402216682341, 7.75842963304955327229332692564, 8.146037049670989337122206299194
