| L(s) = 1 | + 5-s + 4·11-s − 2·17-s + 4·19-s + 25-s + 2·29-s + 10·37-s + 10·41-s − 4·43-s − 8·47-s − 7·49-s + 10·53-s + 4·55-s + 4·59-s − 2·61-s + 12·67-s + 8·71-s − 10·73-s − 12·83-s − 2·85-s − 6·89-s + 4·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.64·37-s + 1.56·41-s − 0.609·43-s − 1.16·47-s − 49-s + 1.37·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.17·73-s − 1.31·83-s − 0.216·85-s − 0.635·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.588415670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.588415670\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.51792563103727, −13.06953191772496, −12.69760905932696, −12.04420399718695, −11.44370895307799, −11.38533248703680, −10.67547498992673, −9.994324114988996, −9.590991870718778, −9.313825984781409, −8.686673440201001, −8.164132619428376, −7.647164533182303, −6.918752232927071, −6.628668180567322, −6.067965110213837, −5.528385339307393, −4.972156553749760, −4.237557802250869, −3.960479667882927, −3.090968900084439, −2.640349335335741, −1.855016099429522, −1.233045137830793, −0.6224395632235109,
0.6224395632235109, 1.233045137830793, 1.855016099429522, 2.640349335335741, 3.090968900084439, 3.960479667882927, 4.237557802250869, 4.972156553749760, 5.528385339307393, 6.067965110213837, 6.628668180567322, 6.918752232927071, 7.647164533182303, 8.164132619428376, 8.686673440201001, 9.313825984781409, 9.590991870718778, 9.994324114988996, 10.67547498992673, 11.38533248703680, 11.44370895307799, 12.04420399718695, 12.69760905932696, 13.06953191772496, 13.51792563103727
