| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 2·5-s + 2·6-s − 5·7-s − 4·8-s + 4·10-s − 2·11-s − 2·12-s − 5·13-s + 10·14-s + 2·15-s + 8·16-s − 2·17-s − 4·20-s + 5·21-s + 4·22-s − 6·23-s + 4·24-s + 3·25-s + 10·26-s + 27-s − 10·28-s + 4·29-s − 4·30-s − 14·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s − 1.88·7-s − 1.41·8-s + 1.26·10-s − 0.603·11-s − 0.577·12-s − 1.38·13-s + 2.67·14-s + 0.516·15-s + 2·16-s − 0.485·17-s − 0.894·20-s + 1.09·21-s + 0.852·22-s − 1.25·23-s + 0.816·24-s + 3/5·25-s + 1.96·26-s + 0.192·27-s − 1.88·28-s + 0.742·29-s − 0.730·30-s − 2.51·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−12.22903265456050207221478858935, −11.83324871917768837763056319596, −11.04285154762065123826130800326, −10.86433363904438056000190268925, −9.917225851315903577146784462859, −9.820408264263129384923454345892, −9.487848919369454543324719087155, −8.874883812046108081362344593792, −8.161078769068842431841901569056, −7.88490645501255769625221699026, −7.17698900774578246291183003390, −6.48950908792604487179936868155, −6.40863876029563659747050316072, −5.44591239539140253733956030522, −4.85195401693137763199433100987, −3.60602569555253128313913909954, −3.25549513136558277618457528319, −2.28099371756223378377514074296, 0, 0,
2.28099371756223378377514074296, 3.25549513136558277618457528319, 3.60602569555253128313913909954, 4.85195401693137763199433100987, 5.44591239539140253733956030522, 6.40863876029563659747050316072, 6.48950908792604487179936868155, 7.17698900774578246291183003390, 7.88490645501255769625221699026, 8.161078769068842431841901569056, 8.874883812046108081362344593792, 9.487848919369454543324719087155, 9.820408264263129384923454345892, 9.917225851315903577146784462859, 10.86433363904438056000190268925, 11.04285154762065123826130800326, 11.83324871917768837763056319596, 12.22903265456050207221478858935
