L(s) = 1 | + (0.5 − 0.866i)3-s + (3 − 1.73i)5-s + (0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (3 + 1.73i)11-s − 5.19i·13-s − 3.46i·15-s + (−6 − 3.46i)17-s + (3.5 + 6.06i)19-s + (−2 − 1.73i)21-s + (3.5 − 6.06i)25-s − 0.999·27-s + (−2.5 + 4.33i)31-s + (3 − 1.73i)33-s + (−3 − 8.66i)35-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (1.34 − 0.774i)5-s + (0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (0.904 + 0.522i)11-s − 1.44i·13-s − 0.894i·15-s + (−1.45 − 0.840i)17-s + (0.802 + 1.39i)19-s + (−0.436 − 0.377i)21-s + (0.700 − 1.21i)25-s − 0.192·27-s + (−0.449 + 0.777i)31-s + (0.522 − 0.301i)33-s + (−0.507 − 1.46i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.408643619\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.408643619\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (6 + 3.46i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 0.866i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (13.5 - 7.79i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \) |
| 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
−9.502575621615633727792216860664, −8.631588017897224189956207000244, −7.78469038409165189007121411511, −6.94866409447265996098729505450, −6.12087609622951812353079906263, −5.24621234750541609240667861761, −4.35739821575581606025693989676, −3.10545492636266755888175225462, −1.80501360781606946745887743509, −1.00834241934176388396059998862,
1.92350600600006433473941044870, 2.45977269823779518765667089920, 3.72158531733819014535825401804, 4.77742325447730044326959309581, 5.80905255413100459293293919254, 6.43165242562004236621811930512, 7.13177564038868560636796121004, 8.711801670108125335304987782737, 9.088715317658547562817399655190, 9.537100351246817620864385934229