| L(s) = 1 | + 2-s + 1.04·3-s + 4-s + 0.635·5-s + 1.04·6-s + 8-s − 1.90·9-s + 0.635·10-s − 6.16·11-s + 1.04·12-s − 3.50·13-s + 0.664·15-s + 16-s − 4.49·17-s − 1.90·18-s + 2.66·19-s + 0.635·20-s − 6.16·22-s + 0.452·23-s + 1.04·24-s − 4.59·25-s − 3.50·26-s − 5.12·27-s + 29-s + 0.664·30-s − 8.57·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.603·3-s + 0.5·4-s + 0.284·5-s + 0.426·6-s + 0.353·8-s − 0.635·9-s + 0.200·10-s − 1.85·11-s + 0.301·12-s − 0.973·13-s + 0.171·15-s + 0.250·16-s − 1.08·17-s − 0.449·18-s + 0.611·19-s + 0.142·20-s − 1.31·22-s + 0.0942·23-s + 0.213·24-s − 0.919·25-s − 0.688·26-s − 0.987·27-s + 0.185·29-s + 0.121·30-s − 1.54·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 - 0.635T + 5T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 23 | \( 1 - 0.452T + 23T^{2} \) |
| 31 | \( 1 + 8.57T + 31T^{2} \) |
| 37 | \( 1 - 3.80T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 + 2.07T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 - 8.43T + 73T^{2} \) |
| 79 | \( 1 + 0.108T + 79T^{2} \) |
| 83 | \( 1 + 0.985T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 2.64T + 97T^{2} \) |
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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
−8.288117597298244424285980093734, −7.59679680683410303393349361505, −7.03087375670337231491535769370, −5.76591172835288043277011200452, −5.39776759759863254610066475449, −4.53403784796927866451706695307, −3.45597454076824555006239620412, −2.55531640420683251630748490223, −2.15170747125300372431730710450, 0,
2.15170747125300372431730710450, 2.55531640420683251630748490223, 3.45597454076824555006239620412, 4.53403784796927866451706695307, 5.39776759759863254610066475449, 5.76591172835288043277011200452, 7.03087375670337231491535769370, 7.59679680683410303393349361505, 8.288117597298244424285980093734
