| L(s) = 1 | + 3-s + 5-s − 2·11-s + 8·13-s + 15-s − 6·17-s − 6·19-s + 8·23-s − 27-s − 4·29-s − 10·31-s − 2·33-s − 2·37-s + 8·39-s + 20·41-s − 8·43-s + 8·47-s − 6·51-s − 4·53-s − 2·55-s − 6·57-s + 8·59-s − 6·61-s + 8·65-s − 12·67-s + 8·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.603·11-s + 2.21·13-s + 0.258·15-s − 1.45·17-s − 1.37·19-s + 1.66·23-s − 0.192·27-s − 0.742·29-s − 1.79·31-s − 0.348·33-s − 0.328·37-s + 1.28·39-s + 3.12·41-s − 1.21·43-s + 1.16·47-s − 0.840·51-s − 0.549·53-s − 0.269·55-s − 0.794·57-s + 1.04·59-s − 0.768·61-s + 0.992·65-s − 1.46·67-s + 0.963·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.923937241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923937241\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.032123755549394024234374162642, −8.732386314251704269522256928874, −8.106732858912908481762957062344, −8.075226010195606770404432832310, −7.33350728398584428779058698463, −7.15139827453886011298603169816, −6.67529767897660997121375452485, −6.15027372937364805742577913082, −6.01635144623902705323283335403, −5.59650834567871191599452129348, −5.10107737253571568880766406330, −4.58494255676543188803966268906, −4.03380453665700381950491571067, −3.94949670010863043063917861447, −3.22172232554499566036538557942, −2.93626869837266583904147871833, −2.19438930573069534441581109368, −1.98760523244162957567858313473, −1.32853070551201040501981019458, −0.52274784577164666050084317066,
0.52274784577164666050084317066, 1.32853070551201040501981019458, 1.98760523244162957567858313473, 2.19438930573069534441581109368, 2.93626869837266583904147871833, 3.22172232554499566036538557942, 3.94949670010863043063917861447, 4.03380453665700381950491571067, 4.58494255676543188803966268906, 5.10107737253571568880766406330, 5.59650834567871191599452129348, 6.01635144623902705323283335403, 6.15027372937364805742577913082, 6.67529767897660997121375452485, 7.15139827453886011298603169816, 7.33350728398584428779058698463, 8.075226010195606770404432832310, 8.106732858912908481762957062344, 8.732386314251704269522256928874, 9.032123755549394024234374162642
