| L(s) = 1 | − 5.04·2-s + 17.4·4-s − 20.1·5-s − 15.4·7-s − 47.7·8-s + 101.·10-s + 26.9·11-s + 77.8·14-s + 101.·16-s − 23.2·17-s + 45.0·19-s − 351.·20-s − 135.·22-s + 142.·23-s + 279.·25-s − 269.·28-s − 2.29·29-s + 37.7·31-s − 128.·32-s + 117.·34-s + 310.·35-s − 313.·37-s − 227.·38-s + 959.·40-s − 5.86·41-s − 360.·43-s + 470.·44-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 2.18·4-s − 1.79·5-s − 0.833·7-s − 2.10·8-s + 3.20·10-s + 0.738·11-s + 1.48·14-s + 1.57·16-s − 0.331·17-s + 0.544·19-s − 3.92·20-s − 1.31·22-s + 1.28·23-s + 2.23·25-s − 1.81·28-s − 0.0146·29-s + 0.218·31-s − 0.708·32-s + 0.591·34-s + 1.49·35-s − 1.39·37-s − 0.970·38-s + 3.79·40-s − 0.0223·41-s − 1.27·43-s + 1.61·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2688808646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2688808646\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.04T + 8T^{2} \) |
| 5 | \( 1 + 20.1T + 125T^{2} \) |
| 7 | \( 1 + 15.4T + 343T^{2} \) |
| 11 | \( 1 - 26.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 23.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.29T + 2.43e4T^{2} \) |
| 31 | \( 1 - 37.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 5.86T + 6.89e4T^{2} \) |
| 43 | \( 1 + 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 209.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 276.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 543.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 205.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 492.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 66.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 317.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 641.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
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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
−9.043574144407916387348949112744, −8.365598157766889667335448473944, −7.70029233511396108277612808876, −6.89191597220384612205881271845, −6.56455219592156002988103451532, −4.88794022407451426610746237814, −3.62210291396119034010953574358, −2.98224324607426310874782425644, −1.38724509636436461059490329200, −0.35539150661988634740455470305,
0.35539150661988634740455470305, 1.38724509636436461059490329200, 2.98224324607426310874782425644, 3.62210291396119034010953574358, 4.88794022407451426610746237814, 6.56455219592156002988103451532, 6.89191597220384612205881271845, 7.70029233511396108277612808876, 8.365598157766889667335448473944, 9.043574144407916387348949112744
