L(s) = 1 | + 1.73i·3-s + (−1.5 − 2.59i)5-s + (−2 − 1.73i)7-s − 2.99·9-s + (−4.5 − 2.59i)11-s + (4.5 − 2.59i)15-s + (−1.5 + 2.59i)17-s + (−1.5 + 0.866i)19-s + (2.99 − 3.46i)21-s + (4.5 − 2.59i)23-s + (−2 + 3.46i)25-s − 5.19i·27-s + (1.5 + 0.866i)31-s + (4.5 − 7.79i)33-s + (−1.5 + 7.79i)35-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (−0.670 − 1.16i)5-s + (−0.755 − 0.654i)7-s − 0.999·9-s + (−1.35 − 0.783i)11-s + (1.16 − 0.670i)15-s + (−0.363 + 0.630i)17-s + (−0.344 + 0.198i)19-s + (0.654 − 0.755i)21-s + (0.938 − 0.541i)23-s + (−0.400 + 0.692i)25-s − 0.999i·27-s + (0.269 + 0.155i)31-s + (0.783 − 1.35i)33-s + (−0.253 + 1.31i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.200496 - 0.374167i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.200496 - 0.374167i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.5 + 2.59i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 2.59i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 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
−10.87179110066611454044049350573, −10.53473846742370792565220207997, −9.280580570965756375617597908928, −8.560535912778259130496403243035, −7.71509209550454139189484929590, −6.13110487055250319504747212186, −5.01980554249672349099764397442, −4.16751471377547432989210386866, −3.06778303700532686479449922194, −0.27910370505178844979233906766,
2.44616229167058933754731263767, 3.16417874537584007200138127684, 5.08693923949794382701179548045, 6.35448889544261147995633234363, 7.13271335894257947375989715937, 7.79018646238769296595474432574, 8.962183920016558485704173350515, 10.16000602608549983971265459692, 11.11589296528930443353598257582, 11.86905209694575460285892198692