| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s − 4·10-s − 4·11-s − 6·13-s − 4·16-s − 6·17-s − 4·20-s − 8·22-s + 12·23-s − 7·25-s − 12·26-s − 8·32-s − 12·34-s + 6·37-s − 8·44-s + 24·46-s + 5·49-s − 14·50-s − 12·52-s + 8·55-s − 20·59-s − 8·64-s + 12·65-s − 24·67-s − 12·68-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s − 1.26·10-s − 1.20·11-s − 1.66·13-s − 16-s − 1.45·17-s − 0.894·20-s − 1.70·22-s + 2.50·23-s − 7/5·25-s − 2.35·26-s − 1.41·32-s − 2.05·34-s + 0.986·37-s − 1.20·44-s + 3.53·46-s + 5/7·49-s − 1.97·50-s − 1.66·52-s + 1.07·55-s − 2.60·59-s − 64-s + 1.48·65-s − 2.93·67-s − 1.45·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391712796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391712796\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.65612740236107695790796582103, −9.838685379266236219076443981276, −9.367058236941436946838956909401, −9.051255729462906967283285122444, −8.677899163356811975775206407300, −7.83881314973015279409779516984, −7.71697366843732888175956990579, −7.13176885768600318265574262979, −7.06301245441252675334246236615, −6.09090916221809287640824532885, −6.05299920579516037395756635088, −5.25323210754301293279961185377, −4.79578636324188286305997361737, −4.59174313634459510366590335032, −4.29353286187484130695075389112, −3.23257123115486763657558912311, −3.22520343473956141973312925965, −2.43280148417257110668833704133, −2.04571798562098403577891944973, −0.41005069647730060555325328768,
0.41005069647730060555325328768, 2.04571798562098403577891944973, 2.43280148417257110668833704133, 3.22520343473956141973312925965, 3.23257123115486763657558912311, 4.29353286187484130695075389112, 4.59174313634459510366590335032, 4.79578636324188286305997361737, 5.25323210754301293279961185377, 6.05299920579516037395756635088, 6.09090916221809287640824532885, 7.06301245441252675334246236615, 7.13176885768600318265574262979, 7.71697366843732888175956990579, 7.83881314973015279409779516984, 8.677899163356811975775206407300, 9.051255729462906967283285122444, 9.367058236941436946838956909401, 9.838685379266236219076443981276, 10.65612740236107695790796582103
