| L(s) = 1 | + 2-s − 3-s + 4-s + 1.28·5-s − 6-s − 0.306·7-s + 8-s + 9-s + 1.28·10-s − 11-s − 12-s − 0.0653·13-s − 0.306·14-s − 1.28·15-s + 16-s − 0.119·17-s + 18-s − 7.96·19-s + 1.28·20-s + 0.306·21-s − 22-s − 1.37·23-s − 24-s − 3.35·25-s − 0.0653·26-s − 27-s − 0.306·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.573·5-s − 0.408·6-s − 0.115·7-s + 0.353·8-s + 0.333·9-s + 0.405·10-s − 0.301·11-s − 0.288·12-s − 0.0181·13-s − 0.0818·14-s − 0.331·15-s + 0.250·16-s − 0.0289·17-s + 0.235·18-s − 1.82·19-s + 0.286·20-s + 0.0668·21-s − 0.213·22-s − 0.286·23-s − 0.204·24-s − 0.671·25-s − 0.0128·26-s − 0.192·27-s − 0.0579·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 0.306T + 7T^{2} \) |
| 13 | \( 1 + 0.0653T + 13T^{2} \) |
| 17 | \( 1 + 0.119T + 17T^{2} \) |
| 19 | \( 1 + 7.96T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 + 7.06T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + 6.46T + 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 67 | \( 1 + 5.22T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 9.21T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 + 5.41T + 89T^{2} \) |
| 97 | \( 1 + 7.22T + 97T^{2} \) |
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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
−8.014940820208875524196979626594, −7.04852532070139280880730385809, −6.44811354033443874277264159320, −5.80426387457918254674153845935, −5.18398055009308513728900298288, −4.32548047180078634043398804189, −3.60148431227892455869481849570, −2.38926441536564147898421262551, −1.69316590072136543261404620486, 0,
1.69316590072136543261404620486, 2.38926441536564147898421262551, 3.60148431227892455869481849570, 4.32548047180078634043398804189, 5.18398055009308513728900298288, 5.80426387457918254674153845935, 6.44811354033443874277264159320, 7.04852532070139280880730385809, 8.014940820208875524196979626594
