L(s) = 1 | + 1.73i·3-s + 4.29·5-s − 2.75i·7-s − 2.99·9-s − 13.7i·11-s + 14.5·13-s + 7.43i·15-s + 22.8·17-s − 16.0i·19-s + 4.77·21-s + 17.1i·23-s − 6.57·25-s − 5.19i·27-s − 21.8·29-s + 38.6i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.858·5-s − 0.393i·7-s − 0.333·9-s − 1.25i·11-s + 1.11·13-s + 0.495i·15-s + 1.34·17-s − 0.844i·19-s + 0.227·21-s + 0.743i·23-s − 0.262·25-s − 0.192i·27-s − 0.754·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.05628\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.05628\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
good | 5 | \( 1 - 4.29T + 25T^{2} \) |
| 7 | \( 1 + 2.75iT - 49T^{2} \) |
| 11 | \( 1 + 13.7iT - 121T^{2} \) |
| 13 | \( 1 - 14.5T + 169T^{2} \) |
| 17 | \( 1 - 22.8T + 289T^{2} \) |
| 19 | \( 1 + 16.0iT - 361T^{2} \) |
| 23 | \( 1 - 17.1iT - 529T^{2} \) |
| 29 | \( 1 + 21.8T + 841T^{2} \) |
| 31 | \( 1 - 38.6iT - 961T^{2} \) |
| 37 | \( 1 - 66.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 23.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 47.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 14.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 65.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 65.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 74.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 122. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 144.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 22.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 122.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 88.3T + 9.40e3T^{2} \) |
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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
−10.96320376534683458486274071570, −10.27008066450660363353974428718, −9.313832192147454381200353453628, −8.576676593134577514414986737327, −7.39944984429357637506618429263, −5.97427377794655409872599685752, −5.54428926048817177702646621152, −3.99476278487736800343135389777, −2.98617778780051879919517157990, −1.12211401472594293858186585503,
1.38939228300423989026835517459, 2.50223000205599719489343386914, 4.09937569932233319517812426338, 5.65268331823332131290599362628, 6.13588881143084743213625986553, 7.42134574772975914519577300323, 8.236487422243013161476163265894, 9.463653432325048172001078337354, 10.01409202005967480316013712153, 11.19283070483248837497063627571