| L(s) = 1 | + 2.55i·2-s − 3.05i·3-s − 4.53·4-s + (1.22 − 1.87i)5-s + 7.82·6-s + 1.35i·7-s − 6.48i·8-s − 6.35·9-s + (4.78 + 3.13i)10-s − 0.813·11-s + 13.8i·12-s − 0.191i·13-s − 3.45·14-s + (−5.72 − 3.74i)15-s + 7.51·16-s − 2.55i·17-s + ⋯ |
| L(s) = 1 | + 1.80i·2-s − 1.76i·3-s − 2.26·4-s + (0.548 − 0.836i)5-s + 3.19·6-s + 0.511i·7-s − 2.29i·8-s − 2.11·9-s + (1.51 + 0.990i)10-s − 0.245·11-s + 4.00i·12-s − 0.0530i·13-s − 0.924·14-s + (−1.47 − 0.967i)15-s + 1.87·16-s − 0.620i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5166091428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5166091428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.22 + 1.87i)T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.55iT - 2T^{2} \) |
| 3 | \( 1 + 3.05iT - 3T^{2} \) |
| 7 | \( 1 - 1.35iT - 7T^{2} \) |
| 11 | \( 1 + 0.813T + 11T^{2} \) |
| 13 | \( 1 + 0.191iT - 13T^{2} \) |
| 17 | \( 1 + 2.55iT - 17T^{2} \) |
| 23 | \( 1 + 5.91iT - 23T^{2} \) |
| 29 | \( 1 - 0.672T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 - 9.62iT - 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 5.13iT - 47T^{2} \) |
| 53 | \( 1 - 3.01iT - 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 - 2.48iT - 67T^{2} \) |
| 71 | \( 1 + 7.28T + 71T^{2} \) |
| 73 | \( 1 + 15.7iT - 73T^{2} \) |
| 79 | \( 1 + 5.99T + 79T^{2} \) |
| 83 | \( 1 + 7.78iT - 83T^{2} \) |
| 89 | \( 1 + 1.91T + 89T^{2} \) |
| 97 | \( 1 - 5.57iT - 97T^{2} \) |
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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
−8.704520745439448402564723686728, −7.976598140255944451673171181327, −7.44371910469161300035207871164, −6.42528667377241261475327663851, −6.22879208189787302525267419297, −5.28133346491336138819413263640, −4.67383260553713229432553744134, −2.79606717332816221440427889537, −1.49135476172725503919693408543, −0.19037830189570089742059543246,
1.80899669832680117280748990444, 2.90066670046466171710888174382, 3.62959670040135690969610992211, 4.09742303749074072464023570772, 5.15789402645542568378915521551, 5.80109796646155119826710010647, 7.32272424808076485262443066974, 8.610532453087610142575868371957, 9.269907777719163942670336397666, 9.858338548153778237925319481694
