| L(s) = 1 | − 0.732·2-s − 1.46·4-s + 0.267·5-s + 7-s + 2.53·8-s − 0.196·10-s − 2.73·13-s − 0.732·14-s + 1.07·16-s − 3.73·17-s + 2.19·19-s − 0.392·20-s − 3.26·23-s − 4.92·25-s + 2·26-s − 1.46·28-s + 1.26·29-s + 7.46·31-s − 5.85·32-s + 2.73·34-s + 0.267·35-s − 9.46·37-s − 1.60·38-s + 0.679·40-s + 4·41-s − 0.464·43-s + 2.39·46-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.732·4-s + 0.119·5-s + 0.377·7-s + 0.896·8-s − 0.0620·10-s − 0.757·13-s − 0.195·14-s + 0.267·16-s − 0.905·17-s + 0.503·19-s − 0.0877·20-s − 0.681·23-s − 0.985·25-s + 0.392·26-s − 0.276·28-s + 0.235·29-s + 1.34·31-s − 1.03·32-s + 0.468·34-s + 0.0452·35-s − 1.55·37-s − 0.260·38-s + 0.107·40-s + 0.624·41-s − 0.0707·43-s + 0.352·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9492007933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9492007933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.732T + 2T^{2} \) |
| 5 | \( 1 - 0.267T + 5T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 0.464T + 43T^{2} \) |
| 47 | \( 1 - 9.19T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 8.19T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 9.12T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 - 0.535T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 3.26T + 97T^{2} \) |
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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
−7.82271697659443327769773234159, −7.49246120516838914451605549742, −6.55895320892468116108277198285, −5.72699612746372057769697907446, −4.95713381034139228636003308805, −4.41796077367318607630670100781, −3.66260622740656559080852838232, −2.50715559904232649040082733645, −1.67925267629047420391650953521, −0.53611777412567401134703813685,
0.53611777412567401134703813685, 1.67925267629047420391650953521, 2.50715559904232649040082733645, 3.66260622740656559080852838232, 4.41796077367318607630670100781, 4.95713381034139228636003308805, 5.72699612746372057769697907446, 6.55895320892468116108277198285, 7.49246120516838914451605549742, 7.82271697659443327769773234159
