| L(s) = 1 | + 3-s + 2·5-s + 2·7-s − 9-s − 8·11-s + 3·13-s + 2·15-s + 2·17-s + 8·19-s + 2·21-s + 2·23-s + 3·25-s − 13·29-s + 7·31-s − 8·33-s + 4·35-s − 6·37-s + 3·39-s − 3·41-s − 10·43-s − 2·45-s + 5·47-s + 6·49-s + 2·51-s + 8·53-s − 16·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s − 1/3·9-s − 2.41·11-s + 0.832·13-s + 0.516·15-s + 0.485·17-s + 1.83·19-s + 0.436·21-s + 0.417·23-s + 3/5·25-s − 2.41·29-s + 1.25·31-s − 1.39·33-s + 0.676·35-s − 0.986·37-s + 0.480·39-s − 0.468·41-s − 1.52·43-s − 0.298·45-s + 0.729·47-s + 6/7·49-s + 0.280·51-s + 1.09·53-s − 2.15·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.999961941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.999961941\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 29 T + 352 T^{2} - 29 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.22479266136955488063613525487, −9.920076549679180655210937480297, −9.445456308088901965682713131926, −9.084272153835367855041594782253, −8.434560251187830677688666690251, −8.282337019996346633165519184144, −7.86840105021195092401430643316, −7.41832271260702522578550728966, −7.06434875404388674376261168701, −6.42466880405790236018053499478, −5.69534474055076709306143378642, −5.38978448874195467279091161608, −5.21122514948191233506734493260, −4.84759225581334813187804428545, −3.60696458038525539065121510260, −3.57529344367571280010444902714, −2.70859664445996752798783022122, −2.38414527944407314457657815859, −1.71934152560981742499742110002, −0.808908830703287407846678614885,
0.808908830703287407846678614885, 1.71934152560981742499742110002, 2.38414527944407314457657815859, 2.70859664445996752798783022122, 3.57529344367571280010444902714, 3.60696458038525539065121510260, 4.84759225581334813187804428545, 5.21122514948191233506734493260, 5.38978448874195467279091161608, 5.69534474055076709306143378642, 6.42466880405790236018053499478, 7.06434875404388674376261168701, 7.41832271260702522578550728966, 7.86840105021195092401430643316, 8.282337019996346633165519184144, 8.434560251187830677688666690251, 9.084272153835367855041594782253, 9.445456308088901965682713131926, 9.920076549679180655210937480297, 10.22479266136955488063613525487
