| L(s) = 1 | − i·2-s − 4-s + 3.46·5-s + (2 − 1.73i)7-s + i·8-s − 3.46i·10-s + 6i·11-s + 1.73i·13-s + (−1.73 − 2i)14-s + 16-s − 1.73·17-s − 6.92i·19-s − 3.46·20-s + 6·22-s − 3i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.54·5-s + (0.755 − 0.654i)7-s + 0.353i·8-s − 1.09i·10-s + 1.80i·11-s + 0.480i·13-s + (−0.462 − 0.534i)14-s + 0.250·16-s − 0.420·17-s − 1.58i·19-s − 0.774·20-s + 1.27·22-s − 0.625i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56819 - 0.716430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56819 - 0.716430i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 8.66T + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 3iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−11.08121340627381757629314675751, −10.29877688104426334052135431195, −9.586658767515454854847101774587, −8.871128503042850255390060382313, −7.39251321746416321140727485722, −6.50200967289606503170204745064, −5.00730869776337642163543849441, −4.44823347395466669575215431044, −2.45996292262262875043062996812, −1.61915126982252147919498589474,
1.65773064102172855830350171016, 3.26702218997399691608563766513, 5.09326916409939077370862112546, 5.80658807229212780804422961734, 6.33805318076580756535377255225, 7.976214258126713115784085842596, 8.599984572855352135130624756224, 9.511582975420566277813295825902, 10.44171597416991846699776687910, 11.38534070428826074083399528369
