| L(s) = 1 | + 4·4-s − 5·7-s − 3·9-s + 12·16-s − 20·28-s − 12·36-s + 20·37-s − 10·43-s + 18·49-s + 15·63-s + 32·64-s + 10·67-s − 8·79-s + 9·81-s + 38·109-s − 60·112-s + 22·121-s + 127-s + 131-s + 137-s + 139-s − 36·144-s + 80·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.88·7-s − 9-s + 3·16-s − 3.77·28-s − 2·36-s + 3.28·37-s − 1.52·43-s + 18/7·49-s + 1.88·63-s + 4·64-s + 1.22·67-s − 0.900·79-s + 81-s + 3.63·109-s − 5.66·112-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3·144-s + 6.57·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.150748434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150748434\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−11.18506512425431643785038834761, −10.80147448811252278177687358691, −10.03660340024899186878097966415, −9.823174331971979521554383397972, −9.692436918001818396013384138378, −8.621549860307797392169778950770, −8.589020569303564902161163192085, −7.64363926778765403064156486729, −7.50316042614573530045073321190, −6.91156973580227517567359219569, −6.28694311484491361433990075268, −6.25688266514178799696251230471, −5.85015050377251278015152898032, −5.23097121185261988116815359526, −4.26828206965667242726157074155, −3.35252138520686902433370894219, −3.24023173503805799586740935190, −2.57306711603478942721687942727, −2.13682601168984960613523377315, −0.836594478994325832410460183014,
0.836594478994325832410460183014, 2.13682601168984960613523377315, 2.57306711603478942721687942727, 3.24023173503805799586740935190, 3.35252138520686902433370894219, 4.26828206965667242726157074155, 5.23097121185261988116815359526, 5.85015050377251278015152898032, 6.25688266514178799696251230471, 6.28694311484491361433990075268, 6.91156973580227517567359219569, 7.50316042614573530045073321190, 7.64363926778765403064156486729, 8.589020569303564902161163192085, 8.621549860307797392169778950770, 9.692436918001818396013384138378, 9.823174331971979521554383397972, 10.03660340024899186878097966415, 10.80147448811252278177687358691, 11.18506512425431643785038834761
