| L(s) = 1 | + 2-s − 1.09·3-s + 4-s − 0.379·5-s − 1.09·6-s − 3.84·7-s + 8-s − 1.80·9-s − 0.379·10-s + 0.261·11-s − 1.09·12-s + 2.14·13-s − 3.84·14-s + 0.415·15-s + 16-s + 7.86·17-s − 1.80·18-s − 1.91·19-s − 0.379·20-s + 4.20·21-s + 0.261·22-s + 23-s − 1.09·24-s − 4.85·25-s + 2.14·26-s + 5.25·27-s − 3.84·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.632·3-s + 0.5·4-s − 0.169·5-s − 0.447·6-s − 1.45·7-s + 0.353·8-s − 0.600·9-s − 0.120·10-s + 0.0788·11-s − 0.316·12-s + 0.594·13-s − 1.02·14-s + 0.107·15-s + 0.250·16-s + 1.90·17-s − 0.424·18-s − 0.439·19-s − 0.0848·20-s + 0.917·21-s + 0.0557·22-s + 0.208·23-s − 0.223·24-s − 0.971·25-s + 0.420·26-s + 1.01·27-s − 0.725·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 1.09T + 3T^{2} \) |
| 5 | \( 1 + 0.379T + 5T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 11 | \( 1 - 0.261T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 7.86T + 17T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 2.25T + 37T^{2} \) |
| 41 | \( 1 - 8.99T + 41T^{2} \) |
| 43 | \( 1 - 0.409T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 1.01T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 - 7.88T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.67184462176592233377296788879, −6.64687097922055067732353841310, −6.17263582389474299967416225403, −5.73995859257441740033101202448, −4.96386712453882607062797563139, −3.91091315577111576074735516174, −3.32441508733929400677120451570, −2.68830678907104315879037451628, −1.23390296826707447294112262249, 0,
1.23390296826707447294112262249, 2.68830678907104315879037451628, 3.32441508733929400677120451570, 3.91091315577111576074735516174, 4.96386712453882607062797563139, 5.73995859257441740033101202448, 6.17263582389474299967416225403, 6.64687097922055067732353841310, 7.67184462176592233377296788879
