| L(s) = 1 | − 4·4-s + 12·9-s + 10·16-s − 8·19-s − 4·25-s − 48·36-s + 32·49-s − 80·61-s − 20·64-s + 32·76-s + 90·81-s + 16·100-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 120·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s − 96·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2·4-s + 4·9-s + 5/2·16-s − 1.83·19-s − 4/5·25-s − 8·36-s + 32/7·49-s − 10.2·61-s − 5/2·64-s + 3.67·76-s + 10·81-s + 8/5·100-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-s − 7.34·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8071807138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8071807138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 7 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 10 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 164 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 182 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.68037594095885346974226115929, −4.64594267254691813195442569194, −4.38951164904599161191131814986, −4.23199634649369632486044375344, −4.21888601047543125117154700003, −4.12495435768697868066564933315, −3.92855912681879237507712714236, −3.70485922896856506949062630135, −3.68170600094730580552962781894, −3.57635907990383575505143292215, −3.54353303507515192840456397248, −3.03806396172307300932222730001, −2.93952964025944464049125240323, −2.79610603177335466150996532040, −2.67776517793726056357387673259, −2.36374246390627645213792062696, −2.15579943594176016556091806640, −2.03533364005633532298823334083, −1.57421912605659168209000045964, −1.54177459750936521018739194175, −1.49828849099031985661421758002, −1.42242707821544713951685268877, −0.854929678458422935899191392745, −0.74000035077477428108076209636, −0.16554038263611980153666059511,
0.16554038263611980153666059511, 0.74000035077477428108076209636, 0.854929678458422935899191392745, 1.42242707821544713951685268877, 1.49828849099031985661421758002, 1.54177459750936521018739194175, 1.57421912605659168209000045964, 2.03533364005633532298823334083, 2.15579943594176016556091806640, 2.36374246390627645213792062696, 2.67776517793726056357387673259, 2.79610603177335466150996532040, 2.93952964025944464049125240323, 3.03806396172307300932222730001, 3.54353303507515192840456397248, 3.57635907990383575505143292215, 3.68170600094730580552962781894, 3.70485922896856506949062630135, 3.92855912681879237507712714236, 4.12495435768697868066564933315, 4.21888601047543125117154700003, 4.23199634649369632486044375344, 4.38951164904599161191131814986, 4.64594267254691813195442569194, 4.68037594095885346974226115929
Plot not available for L-functions of degree greater than 10.
