| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 10·5-s − 4·6-s − 15·9-s + 20·10-s + 4·12-s + 36·13-s − 20·15-s − 4·16-s + 24·17-s + 30·18-s − 12·19-s − 20·20-s + 36·23-s + 75·25-s − 72·26-s − 50·27-s + 48·29-s + 40·30-s + 2·31-s + 8·32-s − 48·34-s − 30·36-s + 24·37-s + 24·38-s + ⋯ |
| L(s) = 1 | − 2-s + 2/3·3-s + 1/2·4-s − 2·5-s − 2/3·6-s − 5/3·9-s + 2·10-s + 1/3·12-s + 2.76·13-s − 4/3·15-s − 1/4·16-s + 1.41·17-s + 5/3·18-s − 0.631·19-s − 20-s + 1.56·23-s + 3·25-s − 2.76·26-s − 1.85·27-s + 1.65·29-s + 4/3·30-s + 2/31·31-s + 1/4·32-s − 1.41·34-s − 5/6·36-s + 0.648·37-s + 0.631·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.087830241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.087830241\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( 1 - 24 T + p^{2} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 217 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2737 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 132 T + 8712 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 84 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 103 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 59 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1847 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 118 T + 6962 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 74 T + 2738 T^{2} - 74 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.15000782193707828524889997927, −11.07982511371556533075678666035, −10.63460459934734560205079778436, −10.15506592584545547514043032707, −9.140785563674600858583411941416, −8.874078284067657034229191919277, −8.562842928216622823985992632025, −8.398633517763880976884612489425, −7.77182098777294645992842759154, −7.63048590619832460904542293791, −6.79024582697427966914545446899, −6.23201103934114681474410370743, −5.78715783494533485008330888027, −4.93161368815040513147552215458, −4.24207831903050033045195396573, −3.44407875719280838114057336306, −3.32678109121232645923293507063, −2.70242763731099665614033533919, −1.18592125511883293463914780605, −0.64400960566822857865478008782,
0.64400960566822857865478008782, 1.18592125511883293463914780605, 2.70242763731099665614033533919, 3.32678109121232645923293507063, 3.44407875719280838114057336306, 4.24207831903050033045195396573, 4.93161368815040513147552215458, 5.78715783494533485008330888027, 6.23201103934114681474410370743, 6.79024582697427966914545446899, 7.63048590619832460904542293791, 7.77182098777294645992842759154, 8.398633517763880976884612489425, 8.562842928216622823985992632025, 8.874078284067657034229191919277, 9.140785563674600858583411941416, 10.15506592584545547514043032707, 10.63460459934734560205079778436, 11.07982511371556533075678666035, 11.15000782193707828524889997927
