| L(s) = 1 | − 2·4-s + 2.23i·5-s + 5.38i·11-s + 4·16-s − 4.47i·20-s − 2.23i·23-s − 5.00·25-s − 5.38i·29-s − 12.0·37-s + 5.38i·41-s − 12.0·43-s − 10.7i·44-s + 7·49-s + 11.1i·53-s − 12.0·55-s + ⋯ |
| L(s) = 1 | − 4-s + 0.999i·5-s + 1.62i·11-s + 16-s − 0.999i·20-s − 0.466i·23-s − 1.00·25-s − 0.999i·29-s − 1.97·37-s + 0.841i·41-s − 1.83·43-s − 1.62i·44-s + 49-s + 1.53i·53-s − 1.62·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4996418101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4996418101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 29 | \( 1 + 5.38iT \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 5.38iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 2.23iT - 23T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 5.38iT - 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 15.6iT - 83T^{2} \) |
| 89 | \( 1 + 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−10.13910534994676241493609506508, −9.364219667902822261002809409808, −8.461168466785031876375582784570, −7.54338755701619960080542557189, −6.89969363321014212174241808316, −5.89743988967450134656050730817, −4.83225155921523667657761638756, −4.12907271947267493859959729610, −3.06491727752146568228042806351, −1.82680416816571196489098261312,
0.22269172729142053465133409986, 1.43833832894140756409834202688, 3.30538821934600939207968873120, 3.99274686870874226041305619069, 5.31977653041580870695401398586, 5.38568888522453096885479601980, 6.72479145089049766712017799806, 7.973200032362912146603035930454, 8.640541769201483444791419317800, 8.951859567458182178738431853278
