| L(s) = 1 | − 0.701·3-s + 5-s − 7-s − 2.50·9-s + 3.37·11-s − 6.46·13-s − 0.701·15-s + 4.49·17-s − 1.76·19-s + 0.701·21-s − 2.72·23-s + 25-s + 3.86·27-s + 3.72·29-s + 7.82·31-s − 2.36·33-s − 35-s + 1.05·37-s + 4.53·39-s + 5.66·41-s − 43-s − 2.50·45-s − 10.8·47-s + 49-s − 3.15·51-s − 5.48·53-s + 3.37·55-s + ⋯ |
| L(s) = 1 | − 0.405·3-s + 0.447·5-s − 0.377·7-s − 0.835·9-s + 1.01·11-s − 1.79·13-s − 0.181·15-s + 1.08·17-s − 0.404·19-s + 0.153·21-s − 0.568·23-s + 0.200·25-s + 0.743·27-s + 0.692·29-s + 1.40·31-s − 0.412·33-s − 0.169·35-s + 0.173·37-s + 0.726·39-s + 0.884·41-s − 0.152·43-s − 0.373·45-s − 1.58·47-s + 0.142·49-s − 0.441·51-s − 0.753·53-s + 0.455·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 0.701T + 3T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.22T + 61T^{2} \) |
| 67 | \( 1 + 5.33T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 + 7.73T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 9.73T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.84845654485358861818144128346, −6.67263141095766885057863460004, −6.47412765206344455799967864296, −5.56304228279064481530729220126, −4.96201623457638361838956417650, −4.14608108198941580843701564946, −3.04708316153391876820129155548, −2.46314198323095154837442547206, −1.23190621099420240584955457161, 0,
1.23190621099420240584955457161, 2.46314198323095154837442547206, 3.04708316153391876820129155548, 4.14608108198941580843701564946, 4.96201623457638361838956417650, 5.56304228279064481530729220126, 6.47412765206344455799967864296, 6.67263141095766885057863460004, 7.84845654485358861818144128346
