| L(s) = 1 | − 5-s + 7-s + 3·11-s + 8·13-s + 4·19-s + 8·23-s + 5·25-s + 6·29-s + 5·31-s − 35-s − 8·37-s − 16·41-s + 12·43-s + 10·47-s − 6·49-s + 9·53-s − 3·55-s − 5·59-s + 10·61-s − 8·65-s − 6·67-s − 20·71-s − 2·73-s + 3·77-s − 11·79-s − 14·83-s − 18·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.904·11-s + 2.21·13-s + 0.917·19-s + 1.66·23-s + 25-s + 1.11·29-s + 0.898·31-s − 0.169·35-s − 1.31·37-s − 2.49·41-s + 1.82·43-s + 1.45·47-s − 6/7·49-s + 1.23·53-s − 0.404·55-s − 0.650·59-s + 1.28·61-s − 0.992·65-s − 0.733·67-s − 2.37·71-s − 0.234·73-s + 0.341·77-s − 1.23·79-s − 1.53·83-s − 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.280267482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.280267482\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−11.16629336353761874326569095686, −10.79456251038982872073303338742, −10.25835875651490948168694898395, −10.00263326156933108535713621077, −9.065068956014744154161589686847, −8.840257192656848746535689843757, −8.573070825867855166115179036695, −8.236893518893041840682346735070, −7.39915350766115575884578249705, −6.96023274190169157949048044455, −6.72930363099288954951121516728, −6.06227841437955689729721380516, −5.50403315614617341918933317591, −5.07057680900019599616935491657, −4.13431469008601160659979963729, −4.12653489596136534809295839707, −3.02037430562676103778859869525, −3.01535134390573336343941041426, −1.32712485904019117575690321372, −1.21089169785677624677076404699,
1.21089169785677624677076404699, 1.32712485904019117575690321372, 3.01535134390573336343941041426, 3.02037430562676103778859869525, 4.12653489596136534809295839707, 4.13431469008601160659979963729, 5.07057680900019599616935491657, 5.50403315614617341918933317591, 6.06227841437955689729721380516, 6.72930363099288954951121516728, 6.96023274190169157949048044455, 7.39915350766115575884578249705, 8.236893518893041840682346735070, 8.573070825867855166115179036695, 8.840257192656848746535689843757, 9.065068956014744154161589686847, 10.00263326156933108535713621077, 10.25835875651490948168694898395, 10.79456251038982872073303338742, 11.16629336353761874326569095686
