| L(s) = 1 | + 2.51e4·7-s + 3.93e4·9-s − 5.24e5·16-s + 1.95e6·19-s + 3.38e6·31-s − 3.07e7·37-s + 3.16e8·49-s + 9.90e8·63-s − 2.25e8·67-s − 5.92e8·73-s + 1.16e9·81-s + 2.57e9·97-s − 4.48e9·109-s − 1.31e10·112-s + 127-s + 131-s + 4.91e10·133-s + 137-s + 139-s − 2.06e10·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.19e9·169-s + 7.68e10·171-s + ⋯ |
| L(s) = 1 | + 3.96·7-s + 2·9-s − 2·16-s + 3.43·19-s + 0.657·31-s − 2.69·37-s + 7.84·49-s + 7.92·63-s − 1.36·67-s − 2.44·73-s + 3·81-s + 2.95·97-s − 3.04·109-s − 7.92·112-s + 13.6·133-s − 4·144-s + 0.678·169-s + 6.87·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(11.64153540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.64153540\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 7197541846 T^{2} + p^{18} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{5} T + p^{9} T^{2} )^{2}( 1 + p^{5} T + p^{9} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + p^{18} T^{4} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12580 T + p^{9} T^{2} )^{2}( 1 + 77549186 T^{2} + p^{18} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + p^{18} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{4} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - 976696 T + p^{9} T^{2} )^{2}( 1 + 308559680858 T^{2} + p^{18} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 1691228 T + p^{9} T^{2} )^{2}( 1 - 50018992173358 T^{2} + p^{18} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 15384490 T + p^{9} T^{2} )^{2}( 1 - 23240947030054 T^{2} + p^{18} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + p^{18} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 16577080 T + p^{9} T^{2} )^{2}( 1 + 16577080 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{18} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{18} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 9487161099916918 T^{2} + p^{18} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 112542320 T + p^{9} T^{2} )^{2}( 1 - 41747295001607494 T^{2} + p^{18} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + p^{18} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + 296368310 T + p^{9} T^{2} )^{2}( 1 - 29908998244279726 T^{2} + p^{18} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 140655567501204338 T^{2} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{18} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + p^{18} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 1288928270 T + p^{9} T^{2} )^{2}( 1 + 140873967896062466 T^{2} + p^{18} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.24831013008342809195294748905, −9.853577493357699989743439120340, −9.035522102556015383151149737403, −9.031903120868612218857466284928, −9.031529992619955957947256484112, −8.027103031228621440076362035772, −7.979950755647399628224679920472, −7.86418433732429073446193464360, −7.14767057512419105334535592940, −7.07808112161910180941955025146, −7.05627427072610286229067225043, −6.11673982519086756575430675585, −5.49323793355386493188315265529, −5.07806127566528073275023163066, −5.00619011885234873087175553450, −4.62007621516604155459927033121, −4.42642986763911421463439278202, −3.87423078433522980694385153418, −3.32087498831037814366707313379, −2.57519145388885662266087942354, −2.02885634896945669922907379171, −1.55305659919034628326724564814, −1.41904854335613630811158838591, −1.20942349770239194415856838442, −0.48627925240060417150767392133,
0.48627925240060417150767392133, 1.20942349770239194415856838442, 1.41904854335613630811158838591, 1.55305659919034628326724564814, 2.02885634896945669922907379171, 2.57519145388885662266087942354, 3.32087498831037814366707313379, 3.87423078433522980694385153418, 4.42642986763911421463439278202, 4.62007621516604155459927033121, 5.00619011885234873087175553450, 5.07806127566528073275023163066, 5.49323793355386493188315265529, 6.11673982519086756575430675585, 7.05627427072610286229067225043, 7.07808112161910180941955025146, 7.14767057512419105334535592940, 7.86418433732429073446193464360, 7.979950755647399628224679920472, 8.027103031228621440076362035772, 9.031529992619955957947256484112, 9.031903120868612218857466284928, 9.035522102556015383151149737403, 9.853577493357699989743439120340, 10.24831013008342809195294748905
