| L(s) = 1 | − 2-s − 2.93·3-s + 4-s + 0.0231·5-s + 2.93·6-s − 4.16·7-s − 8-s + 5.62·9-s − 0.0231·10-s − 0.577·11-s − 2.93·12-s + 4.15·13-s + 4.16·14-s − 0.0680·15-s + 16-s + 5.01·17-s − 5.62·18-s − 4.30·19-s + 0.0231·20-s + 12.2·21-s + 0.577·22-s − 7.61·23-s + 2.93·24-s − 4.99·25-s − 4.15·26-s − 7.70·27-s − 4.16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.69·3-s + 0.5·4-s + 0.0103·5-s + 1.19·6-s − 1.57·7-s − 0.353·8-s + 1.87·9-s − 0.00733·10-s − 0.174·11-s − 0.847·12-s + 1.15·13-s + 1.11·14-s − 0.0175·15-s + 0.250·16-s + 1.21·17-s − 1.32·18-s − 0.986·19-s + 0.00518·20-s + 2.66·21-s + 0.123·22-s − 1.58·23-s + 0.599·24-s − 0.999·25-s − 0.814·26-s − 1.48·27-s − 0.787·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 - 0.0231T + 5T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 + 0.577T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 + 1.79T + 31T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 - 6.87T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 - 9.86T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 + 6.77T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 + 9.52T + 71T^{2} \) |
| 73 | \( 1 - 3.90T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 5.75T + 89T^{2} \) |
| 97 | \( 1 - 6.17T + 97T^{2} \) |
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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
−7.86457182696893666879251031001, −7.32049222009669954740308839120, −6.26807288152260701971886474815, −6.03369207219291227723511806074, −5.62977985621370147376241851306, −4.14770024817532481349867058241, −3.59792102057848256267045088799, −2.19004517397543015373979374540, −0.902673590549635507897566930840, 0,
0.902673590549635507897566930840, 2.19004517397543015373979374540, 3.59792102057848256267045088799, 4.14770024817532481349867058241, 5.62977985621370147376241851306, 6.03369207219291227723511806074, 6.26807288152260701971886474815, 7.32049222009669954740308839120, 7.86457182696893666879251031001
