| L(s) = 1 | − 3-s − 3.56·7-s + 9-s + 3.56·11-s + 13-s − 0.438·17-s − 8.24·19-s + 3.56·21-s + 4.68·23-s − 27-s + 8.24·29-s − 2·31-s − 3.56·33-s + 3.56·37-s − 39-s + 0.438·41-s − 6.24·43-s − 10·47-s + 5.68·49-s + 0.438·51-s + 7.56·53-s + 8.24·57-s + 2.87·59-s − 10.6·61-s − 3.56·63-s + 1.12·67-s − 4.68·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·7-s + 0.333·9-s + 1.07·11-s + 0.277·13-s − 0.106·17-s − 1.89·19-s + 0.777·21-s + 0.976·23-s − 0.192·27-s + 1.53·29-s − 0.359·31-s − 0.619·33-s + 0.585·37-s − 0.160·39-s + 0.0684·41-s − 0.952·43-s − 1.45·47-s + 0.812·49-s + 0.0613·51-s + 1.03·53-s + 1.09·57-s + 0.374·59-s − 1.36·61-s − 0.448·63-s + 0.137·67-s − 0.563·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 17 | \( 1 + 0.438T + 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 - 0.438T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 - 2.87T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 - 5.80T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 7.80T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 5.80T + 97T^{2} \) |
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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
−8.248028442089357266429739074484, −7.02061161414103408353440309145, −6.41813489225411083203969035449, −6.30072273783403590111937005310, −5.09054405683509960779386643549, −4.23917904648596301707728496570, −3.52932014701875324878094094953, −2.54538503513327379040226315354, −1.24917419583378736495346271871, 0,
1.24917419583378736495346271871, 2.54538503513327379040226315354, 3.52932014701875324878094094953, 4.23917904648596301707728496570, 5.09054405683509960779386643549, 6.30072273783403590111937005310, 6.41813489225411083203969035449, 7.02061161414103408353440309145, 8.248028442089357266429739074484
