| L(s) = 1 | + 3·2-s + 3-s + 8·4-s + 18·5-s + 3·6-s + 15·7-s + 45·8-s + 27·9-s + 54·10-s − 48·11-s + 8·12-s + 45·14-s + 18·15-s + 135·16-s − 45·17-s + 81·18-s + 6·19-s + 144·20-s + 15·21-s − 144·22-s + 162·23-s + 45·24-s − 7·25-s + 80·27-s + 120·28-s + 144·29-s + 54·30-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.192·3-s + 4-s + 1.60·5-s + 0.204·6-s + 0.809·7-s + 1.98·8-s + 9-s + 1.70·10-s − 1.31·11-s + 0.192·12-s + 0.859·14-s + 0.309·15-s + 2.10·16-s − 0.642·17-s + 1.06·18-s + 0.0724·19-s + 1.60·20-s + 0.155·21-s − 1.39·22-s + 1.46·23-s + 0.382·24-s − 0.0559·25-s + 0.570·27-s + 0.809·28-s + 0.922·29-s + 0.328·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.668571982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.668571982\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T - 26 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 9 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 15 T - 118 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 48 T + 973 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 45 T - 2888 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T - 6823 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 162 T + 14077 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 144 T - 3653 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 264 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 303 T + 41156 T^{2} - 303 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 192 T - 32057 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 97 T - 70098 T^{2} + 97 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 111 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 414 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 522 T + 67105 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 376 T - 85605 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 36 T - 299467 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 357 T - 230462 T^{2} - 357 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1098 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 830 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 438 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 438 T - 513125 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 852 T - 186769 T^{2} + 852 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.92487119540339450340492987261, −12.47259931478440672729935467258, −11.29348665428651674032558524778, −11.20313200688309415802299414380, −10.43771616252056851052498659990, −10.43738266414971282517613535820, −9.524219806816720275674790216895, −9.349411837209277659567710821598, −8.064598284809918988221014225353, −8.015684715030602850584627632272, −7.04488498164483413830637290130, −6.92687609042333629458542081528, −5.96501195074212143960431944610, −5.33092867123098025423446717616, −5.00639281579602082986976583571, −4.47658821698324116922812953695, −3.59369818419417596579196041306, −2.51438284265894140932485283231, −1.92183126292732341878283082749, −1.38404265891327798468734801706,
1.38404265891327798468734801706, 1.92183126292732341878283082749, 2.51438284265894140932485283231, 3.59369818419417596579196041306, 4.47658821698324116922812953695, 5.00639281579602082986976583571, 5.33092867123098025423446717616, 5.96501195074212143960431944610, 6.92687609042333629458542081528, 7.04488498164483413830637290130, 8.015684715030602850584627632272, 8.064598284809918988221014225353, 9.349411837209277659567710821598, 9.524219806816720275674790216895, 10.43738266414971282517613535820, 10.43771616252056851052498659990, 11.20313200688309415802299414380, 11.29348665428651674032558524778, 12.47259931478440672729935467258, 12.92487119540339450340492987261
