| L(s) = 1 | − 2·3-s + 3·9-s + 8·11-s + 12·17-s + 4·19-s − 2·25-s − 4·27-s − 16·33-s − 20·41-s − 12·43-s + 4·49-s − 24·51-s − 8·57-s + 8·67-s + 32·73-s + 4·75-s + 5·81-s + 32·83-s + 28·89-s − 8·97-s + 24·99-s + 8·107-s − 12·113-s + 26·121-s + 40·123-s + 127-s + 24·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 2.41·11-s + 2.91·17-s + 0.917·19-s − 2/5·25-s − 0.769·27-s − 2.78·33-s − 3.12·41-s − 1.82·43-s + 4/7·49-s − 3.36·51-s − 1.05·57-s + 0.977·67-s + 3.74·73-s + 0.461·75-s + 5/9·81-s + 3.51·83-s + 2.96·89-s − 0.812·97-s + 2.41·99-s + 0.773·107-s − 1.12·113-s + 2.36·121-s + 3.60·123-s + 0.0887·127-s + 2.11·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.901585484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.901585484\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−9.030285325394271519237729699763, −8.435958671936059435057492751632, −7.954005617676551178288923135094, −7.931396215223127360966079874365, −7.25895623666010637445567580839, −6.83599420464659152493558189220, −6.67806811338403373458637637059, −6.20544019607487673640441359670, −6.01108699200988015537896952157, −5.22363521130256737612579690330, −5.10769954754418909808892153844, −5.04161423499886678986819300446, −4.06887742984739526411385386008, −3.71967915242948795311703355467, −3.32815351118757725108372328974, −3.31063943154248586813024128628, −1.87339489422992815585932158497, −1.78983407007564537238033898748, −0.906724278327707418140857629579, −0.819378412879548951261293557368,
0.819378412879548951261293557368, 0.906724278327707418140857629579, 1.78983407007564537238033898748, 1.87339489422992815585932158497, 3.31063943154248586813024128628, 3.32815351118757725108372328974, 3.71967915242948795311703355467, 4.06887742984739526411385386008, 5.04161423499886678986819300446, 5.10769954754418909808892153844, 5.22363521130256737612579690330, 6.01108699200988015537896952157, 6.20544019607487673640441359670, 6.67806811338403373458637637059, 6.83599420464659152493558189220, 7.25895623666010637445567580839, 7.931396215223127360966079874365, 7.954005617676551178288923135094, 8.435958671936059435057492751632, 9.030285325394271519237729699763
