| L(s) = 1 | − 10·25-s + 16·29-s − 40·41-s + 20·49-s + 40·61-s + 24·89-s − 48·101-s − 40·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2·25-s + 2.97·29-s − 6.24·41-s + 20/7·49-s + 5.12·61-s + 2.54·89-s − 4.77·101-s − 3.83·109-s − 0.363·121-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 + 0.923·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9325539793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9325539793\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.81862730369171557358935840590, −6.67009507343241820872957833098, −6.56895241350178818657971136511, −6.34422203322125044833580827475, −5.66412235980262410291366344682, −5.65262436587696134203888218539, −5.50355942463533143707158833452, −5.27728529715644030126361410230, −5.22344490328304362191872505335, −4.80623187152615436896213555662, −4.57501290811589207758319475381, −4.24156084630314207135651686601, −4.17181352203651849841009685803, −3.69317192137638188340017732954, −3.59093306487080246131681157670, −3.57460210232014310424019071743, −2.93279612884887630196428836064, −2.90055613177268380447918844684, −2.46359355507251976451366308267, −2.19099042305534222551153242469, −1.99594864537663159439266731873, −1.63197097223259726290013129297, −1.09406731927388115419791006706, −0.967868677771219234668746747901, −0.19907886645495187645100147373,
0.19907886645495187645100147373, 0.967868677771219234668746747901, 1.09406731927388115419791006706, 1.63197097223259726290013129297, 1.99594864537663159439266731873, 2.19099042305534222551153242469, 2.46359355507251976451366308267, 2.90055613177268380447918844684, 2.93279612884887630196428836064, 3.57460210232014310424019071743, 3.59093306487080246131681157670, 3.69317192137638188340017732954, 4.17181352203651849841009685803, 4.24156084630314207135651686601, 4.57501290811589207758319475381, 4.80623187152615436896213555662, 5.22344490328304362191872505335, 5.27728529715644030126361410230, 5.50355942463533143707158833452, 5.65262436587696134203888218539, 5.66412235980262410291366344682, 6.34422203322125044833580827475, 6.56895241350178818657971136511, 6.67009507343241820872957833098, 6.81862730369171557358935840590
