| L(s) = 1 | + 2·2-s + 0.227·3-s + 4·4-s + 8.31·5-s + 0.455·6-s + 8.08·7-s + 8·8-s − 26.9·9-s + 16.6·10-s − 12.7·11-s + 0.911·12-s − 47.0·13-s + 16.1·14-s + 1.89·15-s + 16·16-s − 31.4·17-s − 53.8·18-s + 19·19-s + 33.2·20-s + 1.84·21-s − 25.5·22-s − 19.0·23-s + 1.82·24-s − 55.8·25-s − 94.0·26-s − 12.3·27-s + 32.3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0438·3-s + 0.5·4-s + 0.743·5-s + 0.0310·6-s + 0.436·7-s + 0.353·8-s − 0.998·9-s + 0.525·10-s − 0.350·11-s + 0.0219·12-s − 1.00·13-s + 0.308·14-s + 0.0326·15-s + 0.250·16-s − 0.448·17-s − 0.705·18-s + 0.229·19-s + 0.371·20-s + 0.0191·21-s − 0.247·22-s − 0.172·23-s + 0.0155·24-s − 0.446·25-s − 0.709·26-s − 0.0876·27-s + 0.218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.933130411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933130411\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 0.227T + 27T^{2} \) |
| 5 | \( 1 - 8.31T + 125T^{2} \) |
| 7 | \( 1 - 8.08T + 343T^{2} \) |
| 11 | \( 1 + 12.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 19.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 293.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 67.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 308.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 108.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 682.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 250.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 317.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 940.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 395.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 975.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 922.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 685.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 211.T + 9.12e5T^{2} \) |
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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
−15.58716924581823310625876813870, −14.38500981277940223423798464016, −13.68656564713630820883903045839, −12.30313414350694990763024205308, −11.12894287555751293338822293772, −9.697880035208362787704356901554, −7.996750157870826228222825275427, −6.20969467878938420561383725721, −4.89012081848617119485394458011, −2.55507133412447475098459378132,
2.55507133412447475098459378132, 4.89012081848617119485394458011, 6.20969467878938420561383725721, 7.996750157870826228222825275427, 9.697880035208362787704356901554, 11.12894287555751293338822293772, 12.30313414350694990763024205308, 13.68656564713630820883903045839, 14.38500981277940223423798464016, 15.58716924581823310625876813870
