| L(s) = 1 | + 4·3-s − 4-s + 6·9-s − 4·12-s − 3·16-s + 6·17-s + 12·23-s − 7·25-s − 4·27-s + 6·29-s − 6·36-s − 16·43-s − 12·48-s − 14·49-s + 24·51-s − 6·53-s + 2·61-s + 7·64-s − 6·68-s + 48·69-s − 28·75-s + 8·79-s − 37·81-s + 24·87-s − 12·92-s + 7·100-s + 6·101-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1/2·4-s + 2·9-s − 1.15·12-s − 3/4·16-s + 1.45·17-s + 2.50·23-s − 7/5·25-s − 0.769·27-s + 1.11·29-s − 36-s − 2.43·43-s − 1.73·48-s − 2·49-s + 3.36·51-s − 0.824·53-s + 0.256·61-s + 7/8·64-s − 0.727·68-s + 5.77·69-s − 3.23·75-s + 0.900·79-s − 4.11·81-s + 2.57·87-s − 1.25·92-s + 7/10·100-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192354221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192354221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
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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
−13.07149786603394019910645113250, −13.01691530904888451970608007968, −12.10428909567156302227165433234, −11.52515822423401509812583099945, −11.14539112755922504548065381680, −10.23881727093134391009984706507, −9.588508745175309869068694658006, −9.583699780833544324547587888885, −8.800662547309621166921013579389, −8.570433187924127333695662515918, −7.999777026071579701241342315621, −7.67417888996219458142284438536, −6.94378083482087772843213542582, −6.25212296055730274651740416579, −5.21823114907427806413664401822, −4.78131403405986186599823458706, −3.70663433293556063675801468526, −3.26403351690720069103565525424, −2.76455864231181104038401891600, −1.71909036867686537100924621887,
1.71909036867686537100924621887, 2.76455864231181104038401891600, 3.26403351690720069103565525424, 3.70663433293556063675801468526, 4.78131403405986186599823458706, 5.21823114907427806413664401822, 6.25212296055730274651740416579, 6.94378083482087772843213542582, 7.67417888996219458142284438536, 7.999777026071579701241342315621, 8.570433187924127333695662515918, 8.800662547309621166921013579389, 9.583699780833544324547587888885, 9.588508745175309869068694658006, 10.23881727093134391009984706507, 11.14539112755922504548065381680, 11.52515822423401509812583099945, 12.10428909567156302227165433234, 13.01691530904888451970608007968, 13.07149786603394019910645113250
