| L(s) = 1 | − 27·3-s + 422·7-s + 486·9-s − 1.97e3·13-s + 1.43e3·19-s − 1.13e4·21-s − 3.12e3·25-s − 6.56e3·27-s + 5.33e4·39-s − 2.24e4·43-s + 9.99e4·49-s − 3.86e4·57-s + 5.69e4·61-s + 2.05e5·63-s + 2.40e3·67-s + 7.81e4·73-s + 8.43e4·75-s + 1.10e5·79-s + 5.90e4·81-s − 8.34e5·91-s + 2.21e5·97-s − 3.81e5·109-s − 9.60e5·117-s + 3.22e5·121-s + 127-s + 6.06e5·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 3.25·7-s + 2·9-s − 3.24·13-s + 0.910·19-s − 5.63·21-s − 25-s − 1.73·27-s + 5.61·39-s − 1.85·43-s + 5.94·49-s − 1.57·57-s + 1.95·61-s + 6.51·63-s + 0.0653·67-s + 1.71·73-s + 1.73·75-s + 1.98·79-s + 81-s − 10.5·91-s + 2.38·97-s − 3.07·109-s − 6.48·117-s + 2·121-s + 5.50e−6·127-s + 3.21·129-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.453056776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453056776\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{3} T + p^{5} T^{2} \) |
| 19 | $C_2$ | \( 1 - 1432 T + p^{5} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p^{5} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 211 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 775 T + p^{5} T^{2} )( 1 + 1202 T + p^{5} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p^{5} T^{2} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p^{5} T^{2} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7601 T + p^{5} T^{2} )( 1 + 7601 T + p^{5} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6661 T + p^{5} T^{2} )( 1 + 6661 T + p^{5} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3352 T + p^{5} T^{2} )( 1 + 19123 T + p^{5} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p^{5} T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 38626 T + p^{5} T^{2} )( 1 - 18301 T + p^{5} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 37939 T + p^{5} T^{2} )( 1 + 35536 T + p^{5} T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 79577 T + p^{5} T^{2} )( 1 + 1450 T + p^{5} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 100564 T + p^{5} T^{2} )( 1 - 9707 T + p^{5} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 177725 T + p^{5} T^{2} )( 1 - 43339 T + p^{5} T^{2} ) \) |
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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
−11.81572293028940315675954869502, −11.22435465124574206024159701883, −10.79966384668333332724298820864, −10.31535116749015738035923597885, −9.754917909425880320874978550690, −9.460615130035924786862790508492, −8.348221382045068166726507182282, −8.013585341632076049263964319706, −7.45488339015839948282773702267, −7.27462864214136711973061060592, −6.57153576803295563915842296002, −5.56970391194177465329090956341, −5.09942979844002645157968873760, −4.99059743222863510914272674506, −4.70851566674511385835326028467, −3.88533977435383482308020907747, −2.32169643251463645229507118625, −1.97886985831684529970345965314, −1.20826470088678706289046460895, −0.42916818962851625735727259525,
0.42916818962851625735727259525, 1.20826470088678706289046460895, 1.97886985831684529970345965314, 2.32169643251463645229507118625, 3.88533977435383482308020907747, 4.70851566674511385835326028467, 4.99059743222863510914272674506, 5.09942979844002645157968873760, 5.56970391194177465329090956341, 6.57153576803295563915842296002, 7.27462864214136711973061060592, 7.45488339015839948282773702267, 8.013585341632076049263964319706, 8.348221382045068166726507182282, 9.460615130035924786862790508492, 9.754917909425880320874978550690, 10.31535116749015738035923597885, 10.79966384668333332724298820864, 11.22435465124574206024159701883, 11.81572293028940315675954869502
