| L(s) = 1 | + 2·2-s + 3·4-s + 6·7-s + 4·8-s + 6·13-s + 12·14-s + 5·16-s − 2·17-s − 2·19-s − 10·23-s + 12·26-s + 18·28-s + 6·29-s + 4·31-s + 6·32-s − 4·34-s − 4·38-s + 12·43-s − 20·46-s + 15·49-s + 18·52-s + 6·53-s + 24·56-s + 12·58-s + 6·59-s + 20·61-s + 8·62-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 2.26·7-s + 1.41·8-s + 1.66·13-s + 3.20·14-s + 5/4·16-s − 0.485·17-s − 0.458·19-s − 2.08·23-s + 2.35·26-s + 3.40·28-s + 1.11·29-s + 0.718·31-s + 1.06·32-s − 0.685·34-s − 0.648·38-s + 1.82·43-s − 2.94·46-s + 15/7·49-s + 2.49·52-s + 0.824·53-s + 3.20·56-s + 1.57·58-s + 0.781·59-s + 2.56·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.97284872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.97284872\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 6 T + 3 p T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 24 T + 4 p T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.916198572879353642813300380482, −7.908348515390718593603931495463, −6.97283679067666865370931078098, −6.90887708408965047979866863337, −6.38626063195253612822281471128, −6.31701654619386153804348470004, −5.62964146597898368826261347087, −5.55814047309776613225682615586, −5.01576708376672632726759356209, −5.00485273216513622461034039567, −4.20467668668607205994053930669, −4.17580433616505669081611166662, −3.78585933314799342505132112432, −3.72324330251560670835138603376, −2.65624352072039710110842186887, −2.51612897798450994033934438120, −2.08197217236639860443911343436, −1.72455973947737535497831794449, −1.10142381880238312606238711056, −0.803923554419488282220457854311,
0.803923554419488282220457854311, 1.10142381880238312606238711056, 1.72455973947737535497831794449, 2.08197217236639860443911343436, 2.51612897798450994033934438120, 2.65624352072039710110842186887, 3.72324330251560670835138603376, 3.78585933314799342505132112432, 4.17580433616505669081611166662, 4.20467668668607205994053930669, 5.00485273216513622461034039567, 5.01576708376672632726759356209, 5.55814047309776613225682615586, 5.62964146597898368826261347087, 6.31701654619386153804348470004, 6.38626063195253612822281471128, 6.90887708408965047979866863337, 6.97283679067666865370931078098, 7.908348515390718593603931495463, 7.916198572879353642813300380482
