| L(s) = 1 | + 2-s + 4-s − 2.79·5-s + 1.31·7-s + 8-s − 2.79·10-s − 0.214·11-s + 4.48·13-s + 1.31·14-s + 16-s − 0.531·17-s − 7.59·19-s − 2.79·20-s − 0.214·22-s + 6.80·23-s + 2.83·25-s + 4.48·26-s + 1.31·28-s − 10.5·29-s + 3.24·31-s + 32-s − 0.531·34-s − 3.68·35-s − 5.24·37-s − 7.59·38-s − 2.79·40-s − 3.41·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.25·5-s + 0.496·7-s + 0.353·8-s − 0.885·10-s − 0.0647·11-s + 1.24·13-s + 0.351·14-s + 0.250·16-s − 0.128·17-s − 1.74·19-s − 0.625·20-s − 0.0458·22-s + 1.41·23-s + 0.567·25-s + 0.880·26-s + 0.248·28-s − 1.96·29-s + 0.582·31-s + 0.176·32-s − 0.0911·34-s − 0.622·35-s − 0.861·37-s − 1.23·38-s − 0.442·40-s − 0.533·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 2.79T + 5T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 + 0.531T + 17T^{2} \) |
| 19 | \( 1 + 7.59T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 47 | \( 1 - 5.87T + 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 - 9.93T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 3.31T + 71T^{2} \) |
| 73 | \( 1 + 2.20T + 73T^{2} \) |
| 79 | \( 1 + 3.11T + 79T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 - 6.38T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.31457974833596977705339536627, −6.70649597707750687693921808061, −6.03547263492669707687776804413, −5.13012131777424900522178356204, −4.55741623891986411118326130612, −3.74312850106312835517748880521, −3.46781522311943570799657038444, −2.27413433931838017972832104776, −1.35197164059294250236551053198, 0,
1.35197164059294250236551053198, 2.27413433931838017972832104776, 3.46781522311943570799657038444, 3.74312850106312835517748880521, 4.55741623891986411118326130612, 5.13012131777424900522178356204, 6.03547263492669707687776804413, 6.70649597707750687693921808061, 7.31457974833596977705339536627
