| L(s) = 1 | − 179.·3-s − 8.71e3·7-s + 1.24e4·9-s − 4.45e4·11-s − 2.14e4·13-s + 3.00e5·17-s − 5.65e5·19-s + 1.56e6·21-s − 9.50e5·23-s + 1.29e6·27-s − 8.03e5·29-s + 1.99e6·31-s + 7.98e6·33-s + 9.53e6·37-s + 3.84e6·39-s − 2.54e7·41-s + 2.32e7·43-s + 3.77e7·47-s + 3.55e7·49-s − 5.38e7·51-s − 4.79e7·53-s + 1.01e8·57-s − 7.00e7·59-s + 1.26e8·61-s − 1.08e8·63-s + 2.66e8·67-s + 1.70e8·69-s + ⋯ |
| L(s) = 1 | − 1.27·3-s − 1.37·7-s + 0.632·9-s − 0.917·11-s − 0.208·13-s + 0.871·17-s − 0.995·19-s + 1.75·21-s − 0.708·23-s + 0.469·27-s − 0.210·29-s + 0.388·31-s + 1.17·33-s + 0.836·37-s + 0.265·39-s − 1.40·41-s + 1.03·43-s + 1.12·47-s + 0.881·49-s − 1.11·51-s − 0.834·53-s + 1.27·57-s − 0.752·59-s + 1.17·61-s − 0.867·63-s + 1.61·67-s + 0.905·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 179.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 8.71e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.14e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.00e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 9.50e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 8.03e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.53e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.54e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.32e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.77e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.79e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.00e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.26e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.66e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.59e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.47e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.66e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.54e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.39e8T + 7.60e17T^{2} \) |
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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
−9.648684152897305961033993643939, −8.339691790867618435398067221862, −7.21822789451180586426055493348, −6.25577380079241500088352465026, −5.73631816861939719017432125391, −4.69874025441917388962084179111, −3.46601439683906503382140920030, −2.35522958215560585129312033719, −0.73043325448786790194759529377, 0,
0.73043325448786790194759529377, 2.35522958215560585129312033719, 3.46601439683906503382140920030, 4.69874025441917388962084179111, 5.73631816861939719017432125391, 6.25577380079241500088352465026, 7.21822789451180586426055493348, 8.339691790867618435398067221862, 9.648684152897305961033993643939
