| L(s) = 1 | + 3·3-s + 9·5-s − 13·7-s + 6·9-s + 15·11-s + 27·15-s + 18·17-s + 18·19-s − 39·21-s + 29·25-s + 9·27-s − 18·29-s + 21·31-s + 45·33-s − 117·35-s − 10·37-s + 148·43-s + 54·45-s + 120·49-s + 54·51-s − 33·53-s + 135·55-s + 54·57-s − 27·59-s + 156·61-s − 78·63-s − 76·67-s + ⋯ |
| L(s) = 1 | + 3-s + 9/5·5-s − 1.85·7-s + 2/3·9-s + 1.36·11-s + 9/5·15-s + 1.05·17-s + 0.947·19-s − 1.85·21-s + 1.15·25-s + 1/3·27-s − 0.620·29-s + 0.677·31-s + 1.36·33-s − 3.34·35-s − 0.270·37-s + 3.44·43-s + 6/5·45-s + 2.44·49-s + 1.05·51-s − 0.622·53-s + 2.45·55-s + 0.947·57-s − 0.457·59-s + 2.55·61-s − 1.23·63-s − 1.13·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.442463264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.442463264\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 13 T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T + 104 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T + 397 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 18 T + 469 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 21 T + 1108 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 10 T - 1269 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3254 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 33 T - 1720 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 27 T + 3724 T^{2} + 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 156 T + 11833 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 76 T + 1287 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 84 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 108 T + 9217 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 43 T - 4392 T^{2} + 43 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 505 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 126 T + 13213 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 15529 T^{2} + p^{4} T^{4} \) |
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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
−11.64891103642191972999255606319, −11.00386401840749677071611147740, −10.13172326461276605405882866830, −10.11464961538529597345457951484, −9.623746694121724989080745404097, −9.377583197239458804223119502884, −8.939471554922477276394065951935, −8.643091511511629784959323555622, −7.48714214818683410704254560186, −7.40956644831861279604947769940, −6.68026245292107641303110333470, −6.14896148297275103036600064630, −5.81947588600138501195542731562, −5.44250933589075319713305125472, −4.21391526464318996618371048976, −3.84658168010773526946828574542, −2.87773296033303320281992838859, −2.85098585410176615202739132520, −1.76510112394175051039959510572, −1.00925810848792942900885055355,
1.00925810848792942900885055355, 1.76510112394175051039959510572, 2.85098585410176615202739132520, 2.87773296033303320281992838859, 3.84658168010773526946828574542, 4.21391526464318996618371048976, 5.44250933589075319713305125472, 5.81947588600138501195542731562, 6.14896148297275103036600064630, 6.68026245292107641303110333470, 7.40956644831861279604947769940, 7.48714214818683410704254560186, 8.643091511511629784959323555622, 8.939471554922477276394065951935, 9.377583197239458804223119502884, 9.623746694121724989080745404097, 10.11464961538529597345457951484, 10.13172326461276605405882866830, 11.00386401840749677071611147740, 11.64891103642191972999255606319
