| L(s) = 1 | − 1.24·3-s − 0.269·5-s + 1.00·7-s − 1.43·9-s + 6.39·11-s − 1.29·13-s + 0.336·15-s − 8.05·17-s + 19-s − 1.25·21-s − 0.229·23-s − 4.92·25-s + 5.54·27-s + 2.49·29-s + 6.82·31-s − 7.98·33-s − 0.270·35-s − 3.53·37-s + 1.61·39-s − 1.41·41-s − 5.00·43-s + 0.388·45-s − 10.1·47-s − 5.99·49-s + 10.0·51-s − 53-s − 1.72·55-s + ⋯ |
| L(s) = 1 | − 0.721·3-s − 0.120·5-s + 0.379·7-s − 0.479·9-s + 1.92·11-s − 0.358·13-s + 0.0870·15-s − 1.95·17-s + 0.229·19-s − 0.273·21-s − 0.0478·23-s − 0.985·25-s + 1.06·27-s + 0.464·29-s + 1.22·31-s − 1.39·33-s − 0.0457·35-s − 0.580·37-s + 0.258·39-s − 0.220·41-s − 0.763·43-s + 0.0578·45-s − 1.48·47-s − 0.856·49-s + 1.40·51-s − 0.137·53-s − 0.232·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + 1.24T + 3T^{2} \) |
| 5 | \( 1 + 0.269T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 - 6.39T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 8.05T + 17T^{2} \) |
| 23 | \( 1 + 0.229T + 23T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 + 5.00T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 - 5.19T + 73T^{2} \) |
| 79 | \( 1 + 6.61T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.408062124408712933427559902031, −7.00837259179124814854633270611, −6.61695313209459299112058690070, −6.00662789596369089181748530739, −4.97907285927610181907222743516, −4.40723497213284331913575101842, −3.56593351570303109749463799704, −2.35564076448946304315112846158, −1.32407457866703743949849189278, 0,
1.32407457866703743949849189278, 2.35564076448946304315112846158, 3.56593351570303109749463799704, 4.40723497213284331913575101842, 4.97907285927610181907222743516, 6.00662789596369089181748530739, 6.61695313209459299112058690070, 7.00837259179124814854633270611, 8.408062124408712933427559902031
