| L(s) = 1 | − 2·7-s − 6·9-s − 4·17-s + 16·23-s − 2·25-s + 16·31-s + 12·41-s + 16·47-s + 3·49-s + 12·63-s + 16·71-s − 12·73-s − 16·79-s + 27·81-s − 12·89-s + 28·97-s − 12·113-s + 8·119-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 2·9-s − 0.970·17-s + 3.33·23-s − 2/5·25-s + 2.87·31-s + 1.87·41-s + 2.33·47-s + 3/7·49-s + 1.51·63-s + 1.89·71-s − 1.40·73-s − 1.80·79-s + 3·81-s − 1.27·89-s + 2.84·97-s − 1.12·113-s + 0.733·119-s − 1.27·121-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 + 1.94·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.775359813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.775359813\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.283207291184141466584345986319, −8.933922207331054765140105443864, −8.849715706945957713738810475150, −8.497708924876449159524637244337, −7.88732880038609807237340711781, −7.58342662965889560029930902122, −6.86970415055069300214451780130, −6.82202198869697479163613700476, −6.20195035941164883316982058710, −5.94105328764709052778105350747, −5.45851803601920151136071006506, −5.10311307022341448100308504821, −4.41284954827906460242568726079, −4.28098626614936484933886845298, −3.30056033995942486401897853493, −3.06394302889213970363524277298, −2.55061108507770883122886202030, −2.41233067412755602077374319604, −1.02142960226701437714894546397, −0.60555351231537538803205726512,
0.60555351231537538803205726512, 1.02142960226701437714894546397, 2.41233067412755602077374319604, 2.55061108507770883122886202030, 3.06394302889213970363524277298, 3.30056033995942486401897853493, 4.28098626614936484933886845298, 4.41284954827906460242568726079, 5.10311307022341448100308504821, 5.45851803601920151136071006506, 5.94105328764709052778105350747, 6.20195035941164883316982058710, 6.82202198869697479163613700476, 6.86970415055069300214451780130, 7.58342662965889560029930902122, 7.88732880038609807237340711781, 8.497708924876449159524637244337, 8.849715706945957713738810475150, 8.933922207331054765140105443864, 9.283207291184141466584345986319
