| L(s) = 1 | + 2.53i·2-s − 4.42·4-s + 3.70i·5-s − 0.957i·7-s − 6.14i·8-s − 9.37·10-s − i·11-s + (−2.10 + 2.92i)13-s + 2.42·14-s + 6.71·16-s − 2.05·17-s − 7.67i·19-s − 16.3i·20-s + 2.53·22-s − 4.20·23-s + ⋯ |
| L(s) = 1 | + 1.79i·2-s − 2.21·4-s + 1.65i·5-s − 0.361i·7-s − 2.17i·8-s − 2.96·10-s − 0.301i·11-s + (−0.584 + 0.811i)13-s + 0.648·14-s + 1.67·16-s − 0.498·17-s − 1.76i·19-s − 3.66i·20-s + 0.540·22-s − 0.877·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3164038893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3164038893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (2.10 - 2.92i)T \) |
| good | 2 | \( 1 - 2.53iT - 2T^{2} \) |
| 5 | \( 1 - 3.70iT - 5T^{2} \) |
| 7 | \( 1 + 0.957iT - 7T^{2} \) |
| 17 | \( 1 + 2.05T + 17T^{2} \) |
| 19 | \( 1 + 7.67iT - 19T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 31 | \( 1 - 10.5iT - 31T^{2} \) |
| 37 | \( 1 + 8.30iT - 37T^{2} \) |
| 41 | \( 1 + 5.23iT - 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 - 1.24iT - 47T^{2} \) |
| 53 | \( 1 + 2.98T + 53T^{2} \) |
| 59 | \( 1 - 12.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.183T + 61T^{2} \) |
| 67 | \( 1 + 1.40iT - 67T^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 3.32iT - 73T^{2} \) |
| 79 | \( 1 + 3.64T + 79T^{2} \) |
| 83 | \( 1 - 3.31iT - 83T^{2} \) |
| 89 | \( 1 + 5.24iT - 89T^{2} \) |
| 97 | \( 1 + 9.82iT - 97T^{2} \) |
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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
−10.25828582046997109406749889620, −9.261058754053588085575563100962, −8.646909321774644028444890654701, −7.44878291197899978716824190386, −7.06194428594705513384631965126, −6.60985604553960261358978671456, −5.70219183587151943711235865065, −4.67073944859539618467304740171, −3.75285564376169552358876588264, −2.49104568977856215985683187109,
0.13353078490291008298519327972, 1.43856165002829971154396557341, 2.26349033956923531548171508388, 3.59649856402171102025399551766, 4.41811817937281605295082989578, 5.13052291979032346082285149145, 5.98823798241428939324507626268, 7.989294049500645109582592842196, 8.257060697989267580954522816088, 9.346940890790045068574185362401
