| L(s) = 1 | + 116·19-s − 116·31-s − 46·49-s − 220·61-s + 416·79-s + 676·109-s + 448·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 578·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 6.10·19-s − 3.74·31-s − 0.938·49-s − 3.60·61-s + 5.26·79-s + 6.20·109-s + 3.70·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.42·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.028150897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.028150897\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 224 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 289 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 400 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 496 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 29 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 398 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2}( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3673 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 368 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6080 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 7609 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8930 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10402 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 12896 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 2590 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17137 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−5.84579790249340369612328131778, −5.50127712683244201829560632186, −5.49869632871953282628849167814, −5.37748410650369803237045275574, −5.11260999848956237517726993262, −4.95399200261780588823317057140, −4.65971829526954425122880450167, −4.52989228666278011735634853935, −4.43525891329179465181496361720, −3.84567857247380859800851345611, −3.63713132764236696865575872058, −3.41982069064425349337395605812, −3.41849478745007897087652028294, −3.26746080290511620756048255632, −3.04470571997464435207270288027, −2.91571370948002101680541703422, −2.47492349906967717879787155460, −1.94665980109686198282336467594, −1.87534173993311096519594261448, −1.81965865617204516442253246701, −1.45498112393510947845205697146, −0.954844957325360050661172165735, −0.790268672960329314109667916405, −0.76976254221068119213133217968, −0.21813711796762481068952988379,
0.21813711796762481068952988379, 0.76976254221068119213133217968, 0.790268672960329314109667916405, 0.954844957325360050661172165735, 1.45498112393510947845205697146, 1.81965865617204516442253246701, 1.87534173993311096519594261448, 1.94665980109686198282336467594, 2.47492349906967717879787155460, 2.91571370948002101680541703422, 3.04470571997464435207270288027, 3.26746080290511620756048255632, 3.41849478745007897087652028294, 3.41982069064425349337395605812, 3.63713132764236696865575872058, 3.84567857247380859800851345611, 4.43525891329179465181496361720, 4.52989228666278011735634853935, 4.65971829526954425122880450167, 4.95399200261780588823317057140, 5.11260999848956237517726993262, 5.37748410650369803237045275574, 5.49869632871953282628849167814, 5.50127712683244201829560632186, 5.84579790249340369612328131778
