| L(s) = 1 | − 2.54i·2-s − 4.49·4-s − 3.49i·5-s + i·7-s + 6.35i·8-s − 8.90·10-s + 0.708i·11-s + (−3.25 − 1.54i)13-s + 2.54·14-s + 7.20·16-s − 7.09·17-s − 0.311i·19-s + 15.6i·20-s + 1.80·22-s + 7.88·23-s + ⋯ |
| L(s) = 1 | − 1.80i·2-s − 2.24·4-s − 1.56i·5-s + 0.377i·7-s + 2.24i·8-s − 2.81·10-s + 0.213i·11-s + (−0.903 − 0.429i)13-s + 0.681·14-s + 1.80·16-s − 1.72·17-s − 0.0714i·19-s + 3.50i·20-s + 0.384·22-s + 1.64·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.455557 + 0.287835i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.455557 + 0.287835i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (3.25 + 1.54i)T \) |
| good | 2 | \( 1 + 2.54iT - 2T^{2} \) |
| 5 | \( 1 + 3.49iT - 5T^{2} \) |
| 11 | \( 1 - 0.708iT - 11T^{2} \) |
| 17 | \( 1 + 7.09T + 17T^{2} \) |
| 19 | \( 1 + 0.311iT - 19T^{2} \) |
| 23 | \( 1 - 7.88T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 - 7.29iT - 31T^{2} \) |
| 37 | \( 1 - 1.41iT - 37T^{2} \) |
| 41 | \( 1 + 11.8iT - 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 + 6.11iT - 47T^{2} \) |
| 53 | \( 1 - 3.72T + 53T^{2} \) |
| 59 | \( 1 - 2.19iT - 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 + 9.17iT - 67T^{2} \) |
| 71 | \( 1 + 0.708iT - 71T^{2} \) |
| 73 | \( 1 - 5.21iT - 73T^{2} \) |
| 79 | \( 1 + 2.78T + 79T^{2} \) |
| 83 | \( 1 - 6.11iT - 83T^{2} \) |
| 89 | \( 1 + 11.1iT - 89T^{2} \) |
| 97 | \( 1 + 7.79iT - 97T^{2} \) |
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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
−9.534335490034554833608576896603, −8.902809328496816531641531379651, −8.555664576226750169366970345842, −7.07613454291346259561961764125, −5.24194871744925229990154578699, −4.90873530851084913430746204464, −3.91373746713489989598581501150, −2.60736575609690931551559552239, −1.61312846008968396303495662732, −0.26434130634099849361488727571,
2.62566912790042944580214907855, 4.00059757513308880187187222175, 4.90762006489605681850315338799, 6.08744969766922910747544874051, 6.79739606727227822574197868169, 7.18950193366055775808636631738, 8.002808330528769329267706802437, 9.120424566650464029709399687286, 9.745606533213831873449132776533, 10.89416218357201781469201790348
