| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 4·8-s + 3·9-s + 4·11-s + 6·12-s + 5·16-s + 6·18-s − 2·19-s + 8·22-s + 4·23-s + 8·24-s − 2·25-s + 4·27-s + 4·29-s − 8·31-s + 6·32-s + 8·33-s + 9·36-s + 4·37-s − 4·38-s + 4·41-s + 16·43-s + 12·44-s + 8·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.41·8-s + 9-s + 1.20·11-s + 1.73·12-s + 5/4·16-s + 1.41·18-s − 0.458·19-s + 1.70·22-s + 0.834·23-s + 1.63·24-s − 2/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s + 1.06·32-s + 1.39·33-s + 3/2·36-s + 0.657·37-s − 0.648·38-s + 0.624·41-s + 2.43·43-s + 1.80·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.09555249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.09555249\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 186 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.115640302793372083477420152088, −7.982784846887823407557650211915, −7.44930798068353424461146170199, −7.26144903926484608163158747612, −6.80379265596786717078071620695, −6.62718645176683169687194761676, −6.02118053957939202579059258839, −5.95377953125461432720214886114, −5.27979585388743941557263992936, −5.07747549851032221009240718688, −4.46211429403156860998865003224, −4.21938683447945886842154960544, −3.74025073656473530275038430221, −3.72389786789076681912868110037, −3.14852122178205003756549848141, −2.62372392554141827030980023453, −2.21776092294904189227348202467, −2.05570968753705216845594622779, −1.16217364667712924837353049275, −0.877555706878572268348603880081,
0.877555706878572268348603880081, 1.16217364667712924837353049275, 2.05570968753705216845594622779, 2.21776092294904189227348202467, 2.62372392554141827030980023453, 3.14852122178205003756549848141, 3.72389786789076681912868110037, 3.74025073656473530275038430221, 4.21938683447945886842154960544, 4.46211429403156860998865003224, 5.07747549851032221009240718688, 5.27979585388743941557263992936, 5.95377953125461432720214886114, 6.02118053957939202579059258839, 6.62718645176683169687194761676, 6.80379265596786717078071620695, 7.26144903926484608163158747612, 7.44930798068353424461146170199, 7.982784846887823407557650211915, 8.115640302793372083477420152088
