| L(s) = 1 | + 2·4-s + 2·7-s + 4·13-s + 6·17-s + 4·19-s − 12·23-s − 4·25-s + 4·28-s + 6·29-s + 16·31-s − 14·37-s − 6·41-s + 4·43-s + 24·47-s + 3·49-s + 8·52-s + 6·53-s + 4·61-s − 8·64-s + 4·67-s + 12·68-s − 12·71-s − 8·73-s + 8·76-s − 2·79-s − 12·83-s + 8·91-s + ⋯ |
| L(s) = 1 | + 4-s + 0.755·7-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 2.50·23-s − 4/5·25-s + 0.755·28-s + 1.11·29-s + 2.87·31-s − 2.30·37-s − 0.937·41-s + 0.609·43-s + 3.50·47-s + 3/7·49-s + 1.10·52-s + 0.824·53-s + 0.512·61-s − 64-s + 0.488·67-s + 1.45·68-s − 1.42·71-s − 0.936·73-s + 0.917·76-s − 0.225·79-s − 1.31·83-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64016001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64016001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.461122381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.461122381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 127 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 172 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 135 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.011752777519427564079207414797, −7.59142499423268438437091792175, −7.43587425908044755501057476873, −6.93245116135414882076674201949, −6.73242015990030196977210017948, −6.03547288909840762799806781183, −5.99581673424060231592990485831, −5.83777342285728021511503054030, −5.13824239300944879145112804146, −5.08630175609699120559103937533, −4.21167849494724243594894881229, −4.16772098887322074841561776458, −3.78876775369904540773722427496, −3.24591463955147718577444817011, −2.80222178944816551373719985819, −2.46312038230056082614469831264, −2.03780219137573842424714449543, −1.36281015229573483665711493292, −1.28266928430766931766973504102, −0.55150585258363356747223626786,
0.55150585258363356747223626786, 1.28266928430766931766973504102, 1.36281015229573483665711493292, 2.03780219137573842424714449543, 2.46312038230056082614469831264, 2.80222178944816551373719985819, 3.24591463955147718577444817011, 3.78876775369904540773722427496, 4.16772098887322074841561776458, 4.21167849494724243594894881229, 5.08630175609699120559103937533, 5.13824239300944879145112804146, 5.83777342285728021511503054030, 5.99581673424060231592990485831, 6.03547288909840762799806781183, 6.73242015990030196977210017948, 6.93245116135414882076674201949, 7.43587425908044755501057476873, 7.59142499423268438437091792175, 8.011752777519427564079207414797
