| L(s) = 1 | + 2·3-s − 5-s + 3·9-s − 12·13-s − 2·15-s + 25-s + 4·27-s − 12·37-s − 24·39-s + 20·41-s − 8·43-s − 3·45-s + 2·49-s + 20·53-s + 12·65-s − 8·67-s + 2·75-s + 32·79-s + 5·81-s + 24·83-s + 4·89-s + 8·107-s − 24·111-s − 36·117-s − 22·121-s + 40·123-s − 125-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 9-s − 3.32·13-s − 0.516·15-s + 1/5·25-s + 0.769·27-s − 1.97·37-s − 3.84·39-s + 3.12·41-s − 1.21·43-s − 0.447·45-s + 2/7·49-s + 2.74·53-s + 1.48·65-s − 0.977·67-s + 0.230·75-s + 3.60·79-s + 5/9·81-s + 2.63·83-s + 0.423·89-s + 0.773·107-s − 2.27·111-s − 3.32·117-s − 2·121-s + 3.60·123-s − 0.0894·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.960360774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.960360774\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.303895764062546122836103468548, −7.929044973464791963060046214031, −7.47733110912306642666316245595, −7.34594365176835478558310775458, −6.84540072511643881723777212979, −6.35824384462791146836852048356, −5.28897888504504330389281711769, −5.24425276810078724988361684693, −4.57846166641473825300156973670, −4.13728859524865050162716547657, −3.51779285053220397913079365279, −2.91626142161817301946825099474, −2.20072377464036840364167903651, −2.19180563382997996757520036314, −0.64210289953953935694791606107,
0.64210289953953935694791606107, 2.19180563382997996757520036314, 2.20072377464036840364167903651, 2.91626142161817301946825099474, 3.51779285053220397913079365279, 4.13728859524865050162716547657, 4.57846166641473825300156973670, 5.24425276810078724988361684693, 5.28897888504504330389281711769, 6.35824384462791146836852048356, 6.84540072511643881723777212979, 7.34594365176835478558310775458, 7.47733110912306642666316245595, 7.929044973464791963060046214031, 8.303895764062546122836103468548
