| L(s) = 1 | + 96·41-s + 88·61-s − 2·81-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 14.9·41-s + 11.2·61-s − 2/9·81-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.85123707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.85123707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 71 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 337 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 958 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 12 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 3527 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 1054 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 2254 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 11 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 7753 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 1054 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 17663 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−4.21258075528658638118531203391, −4.13653175918738664482945911220, −3.81923767350194196405603185883, −3.77152062493115846980239592801, −3.62598641613288493278933966277, −3.58230368844344379717891170902, −3.48571804033745000388437301910, −3.46883817537635679804410723982, −3.09213002052733937205047782522, −2.70403317685920330871428210151, −2.54466728471039211379512450932, −2.51426468931372058022355830319, −2.45482088691648310309225966152, −2.42809511043274122446468802173, −2.40789152569434226535480555400, −2.34035526837349029197688099121, −2.23729284972511720940773630141, −1.70269227677159868361023466373, −1.50391071924271547806350914087, −1.15850060936628830915979270958, −1.10310520173948835535628741585, −0.835914117626861800489082111688, −0.74250877331787303266006859161, −0.62205812008630598514591167991, −0.61171083605921929792897223406,
0.61171083605921929792897223406, 0.62205812008630598514591167991, 0.74250877331787303266006859161, 0.835914117626861800489082111688, 1.10310520173948835535628741585, 1.15850060936628830915979270958, 1.50391071924271547806350914087, 1.70269227677159868361023466373, 2.23729284972511720940773630141, 2.34035526837349029197688099121, 2.40789152569434226535480555400, 2.42809511043274122446468802173, 2.45482088691648310309225966152, 2.51426468931372058022355830319, 2.54466728471039211379512450932, 2.70403317685920330871428210151, 3.09213002052733937205047782522, 3.46883817537635679804410723982, 3.48571804033745000388437301910, 3.58230368844344379717891170902, 3.62598641613288493278933966277, 3.77152062493115846980239592801, 3.81923767350194196405603185883, 4.13653175918738664482945911220, 4.21258075528658638118531203391
Plot not available for L-functions of degree greater than 10.
