L(s) = 1 | + (−0.5 − 0.866i)3-s + (1.5 + 0.866i)5-s + (0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (1.5 − 0.866i)11-s − 1.73i·15-s + (3 − 1.73i)17-s + (1 − 1.73i)19-s + (−2.5 + 0.866i)21-s + (−1 − 1.73i)25-s + 0.999·27-s + 9·29-s + (−2.5 − 4.33i)31-s + (−1.5 − 0.866i)33-s + (3 − 3.46i)35-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.670 + 0.387i)5-s + (0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.452 − 0.261i)11-s − 0.447i·15-s + (0.727 − 0.420i)17-s + (0.229 − 0.397i)19-s + (−0.545 + 0.188i)21-s + (−0.200 − 0.346i)25-s + 0.192·27-s + 1.67·29-s + (−0.449 − 0.777i)31-s + (−0.261 − 0.150i)33-s + (0.507 − 0.585i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26437 - 0.564994i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26437 - 0.564994i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-3 + 1.73i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12 - 6.92i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + (3 + 1.73i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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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
−11.44745381110301718548587896288, −10.46377590517617138320775756959, −9.819315281793336142955598128581, −8.531191304456363468334837613519, −7.42819210345826005041097765573, −6.65753218298887524195105826938, −5.68545939867883456956698803485, −4.38546512750434789043718867075, −2.86885285692459854617263203460, −1.19226971877493348484139312883,
1.76568093759239594602061038249, 3.41515829207316476287489313638, 4.91742076453252468906770468423, 5.63013305515793625995455845003, 6.62133921340766585861987217264, 8.138714065745209770547273243674, 9.031441912788082727672425083054, 9.776979601332275083691057184293, 10.63515159240329913830380747356, 11.86955287888147306674300484446