| L(s) = 1 | − 3-s − 5-s − 2.11·7-s + 9-s − 1.80·11-s + 2.27·13-s + 15-s − 7.04·17-s + 7.64·19-s + 2.11·21-s − 8.99·23-s + 25-s − 27-s + 4.88·29-s − 4.18·31-s + 1.80·33-s + 2.11·35-s + 2.77·37-s − 2.27·39-s − 2.86·41-s − 0.432·43-s − 45-s − 5.20·47-s − 2.54·49-s + 7.04·51-s − 1.87·53-s + 1.80·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.797·7-s + 0.333·9-s − 0.543·11-s + 0.630·13-s + 0.258·15-s − 1.70·17-s + 1.75·19-s + 0.460·21-s − 1.87·23-s + 0.200·25-s − 0.192·27-s + 0.907·29-s − 0.751·31-s + 0.314·33-s + 0.356·35-s + 0.456·37-s − 0.363·39-s − 0.447·41-s − 0.0660·43-s − 0.149·45-s − 0.759·47-s − 0.363·49-s + 0.986·51-s − 0.257·53-s + 0.243·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8049208310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8049208310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 - 2.27T + 13T^{2} \) |
| 17 | \( 1 + 7.04T + 17T^{2} \) |
| 19 | \( 1 - 7.64T + 19T^{2} \) |
| 23 | \( 1 + 8.99T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 + 0.432T + 43T^{2} \) |
| 47 | \( 1 + 5.20T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 - 3.24T + 59T^{2} \) |
| 61 | \( 1 - 2.12T + 61T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 - 9.69T + 73T^{2} \) |
| 79 | \( 1 - 5.75T + 79T^{2} \) |
| 83 | \( 1 + 7.66T + 83T^{2} \) |
| 89 | \( 1 + 0.339T + 89T^{2} \) |
| 97 | \( 1 + 4.25T + 97T^{2} \) |
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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
−8.348851914757471511851624089782, −7.71500798346576630968990042818, −6.83153910673698436475814842483, −6.31375869973083669519313558034, −5.52705945995672681775534844537, −4.68778901643079554804261162669, −3.85842850895220614573597254189, −3.07845985569576340882543261462, −1.93514291458683512039310063849, −0.51188023935418826187233209036,
0.51188023935418826187233209036, 1.93514291458683512039310063849, 3.07845985569576340882543261462, 3.85842850895220614573597254189, 4.68778901643079554804261162669, 5.52705945995672681775534844537, 6.31375869973083669519313558034, 6.83153910673698436475814842483, 7.71500798346576630968990042818, 8.348851914757471511851624089782
