| L(s) = 1 | + 4.62·3-s − 17.8·5-s + 13.8·7-s − 5.59·9-s − 2.18·11-s + 46.3·13-s − 82.7·15-s − 18.6·17-s − 122.·19-s + 64.1·21-s + 134.·23-s + 194.·25-s − 150.·27-s + 84.5·29-s − 31.5·31-s − 10.1·33-s − 248.·35-s + 126.·37-s + 214.·39-s + 210.·41-s − 168.·43-s + 100.·45-s + 182.·47-s − 150.·49-s − 86.2·51-s + 37.0·53-s + 39.1·55-s + ⋯ |
| L(s) = 1 | + 0.890·3-s − 1.59·5-s + 0.749·7-s − 0.207·9-s − 0.0599·11-s + 0.989·13-s − 1.42·15-s − 0.266·17-s − 1.47·19-s + 0.667·21-s + 1.21·23-s + 1.55·25-s − 1.07·27-s + 0.541·29-s − 0.182·31-s − 0.0533·33-s − 1.19·35-s + 0.560·37-s + 0.880·39-s + 0.801·41-s − 0.598·43-s + 0.331·45-s + 0.567·47-s − 0.438·49-s − 0.236·51-s + 0.0958·53-s + 0.0959·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.046303324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.046303324\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4.62T + 27T^{2} \) |
| 5 | \( 1 + 17.8T + 125T^{2} \) |
| 7 | \( 1 - 13.8T + 343T^{2} \) |
| 11 | \( 1 + 2.18T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 84.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 37.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 624.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 246.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 129.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 348.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 299.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 943.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 443.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−9.178066843917818205370677078977, −8.463710104634004924104836469355, −8.177830283690297228415335920867, −7.33284869981422888796265979773, −6.32754477513108282123565979602, −4.94030930157563842186892726486, −4.06956746066854671690371895257, −3.37349326213252338413056745894, −2.24329270540233609938183858733, −0.72215083035860880544702619973,
0.72215083035860880544702619973, 2.24329270540233609938183858733, 3.37349326213252338413056745894, 4.06956746066854671690371895257, 4.94030930157563842186892726486, 6.32754477513108282123565979602, 7.33284869981422888796265979773, 8.177830283690297228415335920867, 8.463710104634004924104836469355, 9.178066843917818205370677078977
