| L(s) = 1 | − 4·3-s − 2·4-s − 4·7-s + 10·9-s − 4·11-s + 8·12-s + 3·16-s + 16·21-s − 2·25-s − 20·27-s + 8·28-s + 16·33-s − 20·36-s + 2·37-s + 14·41-s + 8·44-s − 12·48-s − 18·49-s − 8·53-s − 40·63-s − 4·64-s + 4·67-s − 24·71-s + 18·73-s + 8·75-s + 16·77-s + 35·81-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 4-s − 1.51·7-s + 10/3·9-s − 1.20·11-s + 2.30·12-s + 3/4·16-s + 3.49·21-s − 2/5·25-s − 3.84·27-s + 1.51·28-s + 2.78·33-s − 3.33·36-s + 0.328·37-s + 2.18·41-s + 1.20·44-s − 1.73·48-s − 2.57·49-s − 1.09·53-s − 5.03·63-s − 1/2·64-s + 0.488·67-s − 2.84·71-s + 2.10·73-s + 0.923·75-s + 1.82·77-s + 35/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2519064147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2519064147\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 15 T^{2} + 376 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 15 T^{2} - 116 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 404 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 55 T^{2} + 1544 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 63 T^{2} + 2020 T^{4} - 63 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 7 T + 76 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 75 T^{2} + 2896 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 8606 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 63 T^{2} + 3160 T^{4} - 63 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 9 T + 148 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 152 T^{2} + 15630 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 5 T + 154 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 13 T^{2} + 2580 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 267 T^{2} + 33556 T^{4} - 267 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.77229265427398682634036357251, −6.76973644957434525780156071947, −6.52132167812112477064575626731, −6.30683605479659631785898456153, −6.30067219957520895055800718777, −5.70299252086931512555366913503, −5.60871968477146653844467978363, −5.59585216326895650393756781464, −5.44389636417750603702632394959, −4.96127408925626484358602737948, −4.81837070123632901181594673101, −4.60310440351507024199118550179, −4.30405368165470605622528409320, −4.22812394778794173437952286527, −3.89515936692871272690321920833, −3.51964581800776394351998655455, −3.19455780887877874026154625075, −3.05647866067484645448873503188, −2.80517256711216622281089036468, −2.16089664843986875794577008380, −2.06713684449809413188029944956, −1.29908672836102407920022994857, −1.24285261405512125774224792809, −0.39298030441494478664562350435, −0.34332338696476580850486373978,
0.34332338696476580850486373978, 0.39298030441494478664562350435, 1.24285261405512125774224792809, 1.29908672836102407920022994857, 2.06713684449809413188029944956, 2.16089664843986875794577008380, 2.80517256711216622281089036468, 3.05647866067484645448873503188, 3.19455780887877874026154625075, 3.51964581800776394351998655455, 3.89515936692871272690321920833, 4.22812394778794173437952286527, 4.30405368165470605622528409320, 4.60310440351507024199118550179, 4.81837070123632901181594673101, 4.96127408925626484358602737948, 5.44389636417750603702632394959, 5.59585216326895650393756781464, 5.60871968477146653844467978363, 5.70299252086931512555366913503, 6.30067219957520895055800718777, 6.30683605479659631785898456153, 6.52132167812112477064575626731, 6.76973644957434525780156071947, 6.77229265427398682634036357251
