| L(s) = 1 | + 5.83·5-s + 20.2·7-s − 59.1·11-s + 21.2·13-s + 93.2·17-s + 69.4·19-s − 34.2·23-s − 90.8·25-s − 29·29-s + 155.·31-s + 118.·35-s + 117.·37-s + 325.·41-s − 358.·43-s + 287.·47-s + 67.5·49-s + 496.·53-s − 345.·55-s + 474.·59-s + 136.·61-s + 124.·65-s − 42.5·67-s − 266.·71-s + 735.·73-s − 1.19e3·77-s − 1.18e3·79-s − 962.·83-s + ⋯ |
| L(s) = 1 | + 0.522·5-s + 1.09·7-s − 1.62·11-s + 0.453·13-s + 1.33·17-s + 0.838·19-s − 0.310·23-s − 0.727·25-s − 0.185·29-s + 0.898·31-s + 0.571·35-s + 0.521·37-s + 1.24·41-s − 1.27·43-s + 0.892·47-s + 0.196·49-s + 1.28·53-s − 0.847·55-s + 1.04·59-s + 0.287·61-s + 0.236·65-s − 0.0776·67-s − 0.444·71-s + 1.18·73-s − 1.77·77-s − 1.68·79-s − 1.27·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.657827858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.657827858\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 5 | \( 1 - 5.83T + 125T^{2} \) |
| 7 | \( 1 - 20.2T + 343T^{2} \) |
| 11 | \( 1 + 59.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 34.2T + 1.21e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 358.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 287.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 496.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 474.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 136.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 42.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 266.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 962.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 709.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 413.T + 9.12e5T^{2} \) |
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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
−9.810738909633552001691127195478, −8.494123825771448368232699217889, −7.935092454510516010661103596802, −7.25945873619453039801345328438, −5.74296213022356081136569605283, −5.45887872668972869263496420239, −4.38957262207874274081848292637, −3.08003647612242031262333512766, −2.05180640778616743208619859373, −0.891314169934282493606507807991,
0.891314169934282493606507807991, 2.05180640778616743208619859373, 3.08003647612242031262333512766, 4.38957262207874274081848292637, 5.45887872668972869263496420239, 5.74296213022356081136569605283, 7.25945873619453039801345328438, 7.935092454510516010661103596802, 8.494123825771448368232699217889, 9.810738909633552001691127195478
