| L(s) = 1 | + 2·2-s + 3·4-s + 4·5-s + 2·7-s + 4·8-s + 8·10-s + 6·11-s + 4·14-s + 5·16-s + 8·17-s − 6·19-s + 12·20-s + 12·22-s + 2·23-s + 5·25-s + 6·28-s + 2·29-s + 4·31-s + 6·32-s + 16·34-s + 8·35-s − 14·37-s − 12·38-s + 16·40-s + 2·41-s − 2·43-s + 18·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.78·5-s + 0.755·7-s + 1.41·8-s + 2.52·10-s + 1.80·11-s + 1.06·14-s + 5/4·16-s + 1.94·17-s − 1.37·19-s + 2.68·20-s + 2.55·22-s + 0.417·23-s + 25-s + 1.13·28-s + 0.371·29-s + 0.718·31-s + 1.06·32-s + 2.74·34-s + 1.35·35-s − 2.30·37-s − 1.94·38-s + 2.52·40-s + 0.312·41-s − 0.304·43-s + 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.64546154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.64546154\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 47 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 3 p T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 207 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.777285779634056065292063414770, −8.617049658119660568483925116553, −8.047840735048900369522183051577, −7.84798522162871798466678426427, −6.98057238369366006138231477940, −6.92462666217734711604906139166, −6.42472535630975994459228731397, −6.30846835238485380523293235764, −5.68561993460932933947777542085, −5.52630091215188444648591721997, −5.07401177267264750177802321262, −4.81707916508546336497260291380, −4.12387342501864834954304991178, −3.92345352723761991098957054046, −3.31062698000435634896133731644, −3.01976516942970441412326113851, −2.17333292392493369761009921253, −1.96037030203415569551932536120, −1.44980365953219590467071226800, −1.06160160747160921100454593550,
1.06160160747160921100454593550, 1.44980365953219590467071226800, 1.96037030203415569551932536120, 2.17333292392493369761009921253, 3.01976516942970441412326113851, 3.31062698000435634896133731644, 3.92345352723761991098957054046, 4.12387342501864834954304991178, 4.81707916508546336497260291380, 5.07401177267264750177802321262, 5.52630091215188444648591721997, 5.68561993460932933947777542085, 6.30846835238485380523293235764, 6.42472535630975994459228731397, 6.92462666217734711604906139166, 6.98057238369366006138231477940, 7.84798522162871798466678426427, 8.047840735048900369522183051577, 8.617049658119660568483925116553, 8.777285779634056065292063414770
