| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 5-s + 2·6-s − 4·7-s + 4·8-s + 2·10-s + 3·11-s + 3·12-s + 2·13-s − 8·14-s + 15-s + 5·16-s − 3·17-s + 3·20-s − 4·21-s + 6·22-s + 6·23-s + 4·24-s + 4·26-s − 27-s − 12·28-s + 4·29-s + 2·30-s + 4·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s + 1.41·8-s + 0.632·10-s + 0.904·11-s + 0.866·12-s + 0.554·13-s − 2.13·14-s + 0.258·15-s + 5/4·16-s − 0.727·17-s + 0.670·20-s − 0.872·21-s + 1.27·22-s + 1.25·23-s + 0.816·24-s + 0.784·26-s − 0.192·27-s − 2.26·28-s + 0.742·29-s + 0.365·30-s + 0.718·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.555500076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.555500076\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.09614873018416241204703557101, −10.09525906558232194306946797242, −9.303161367808781556979367266984, −9.199584065793971541431103152329, −8.569516861440567197812080845400, −8.341615241896789218493879833880, −7.42279795918684088903537232660, −7.22425382334881124565689590245, −6.49775466888027641132114124234, −6.43982957829245950124572615205, −6.15496752736418220895976880189, −5.47668710373780380993700138658, −4.93861677558081285453540997981, −4.48950715216738713579792303945, −3.72833159292597274449965840144, −3.66446007861820986766826500574, −2.86405044815947304372246796157, −2.69008970981109171078380846317, −1.86724401278316700695794125397, −0.968926216923291339214874450952,
0.968926216923291339214874450952, 1.86724401278316700695794125397, 2.69008970981109171078380846317, 2.86405044815947304372246796157, 3.66446007861820986766826500574, 3.72833159292597274449965840144, 4.48950715216738713579792303945, 4.93861677558081285453540997981, 5.47668710373780380993700138658, 6.15496752736418220895976880189, 6.43982957829245950124572615205, 6.49775466888027641132114124234, 7.22425382334881124565689590245, 7.42279795918684088903537232660, 8.341615241896789218493879833880, 8.569516861440567197812080845400, 9.199584065793971541431103152329, 9.303161367808781556979367266984, 10.09525906558232194306946797242, 10.09614873018416241204703557101
