| L(s) = 1 | + 7-s + 6·11-s − 6·13-s + 3·17-s − 4·19-s − 2·23-s + 4·29-s − 5·31-s − 10·37-s + 41-s − 9·43-s − 10·47-s − 12·49-s + 5·53-s + 7·59-s − 11·61-s − 16·67-s + 13·71-s + 3·73-s + 6·77-s − 6·79-s − 13·83-s − 2·89-s − 6·91-s + 97-s + 20·101-s + 11·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.80·11-s − 1.66·13-s + 0.727·17-s − 0.917·19-s − 0.417·23-s + 0.742·29-s − 0.898·31-s − 1.64·37-s + 0.156·41-s − 1.37·43-s − 1.45·47-s − 1.71·49-s + 0.686·53-s + 0.911·59-s − 1.40·61-s − 1.95·67-s + 1.54·71-s + 0.351·73-s + 0.683·77-s − 0.675·79-s − 1.42·83-s − 0.211·89-s − 0.628·91-s + 0.101·97-s + 1.99·101-s + 1.08·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 67 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - T + 51 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 95 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 11 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 99 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 91 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 173 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 147 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 207 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 159 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 193 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−7.54726719309340410090036462955, −7.18095392503210580146057814504, −6.79867208305399667236234262571, −6.69843206989267076899089224253, −6.15387871654629026818727070313, −6.06135119365258827593464057962, −5.25821957376228107350716979989, −5.24086959334273367061854008256, −4.64905835335895814375447868681, −4.62890208332682623497980215809, −3.85361036268767644047246528884, −3.85300886583032049669113761785, −3.13709634540880602580262409944, −3.06687510260869426300241099772, −2.20606640864549811066135930900, −2.02221382474457516379643476012, −1.39710677739232928864602318780, −1.25292132931688424660773345651, 0, 0,
1.25292132931688424660773345651, 1.39710677739232928864602318780, 2.02221382474457516379643476012, 2.20606640864549811066135930900, 3.06687510260869426300241099772, 3.13709634540880602580262409944, 3.85300886583032049669113761785, 3.85361036268767644047246528884, 4.62890208332682623497980215809, 4.64905835335895814375447868681, 5.24086959334273367061854008256, 5.25821957376228107350716979989, 6.06135119365258827593464057962, 6.15387871654629026818727070313, 6.69843206989267076899089224253, 6.79867208305399667236234262571, 7.18095392503210580146057814504, 7.54726719309340410090036462955
