L(s) = 1 | + (−1.22 + 1.22i)3-s + 2.44·5-s + (−1 + 2.44i)7-s − 2.99i·9-s + 2.44i·13-s + (−2.99 + 2.99i)15-s + 4.89·17-s − 2.44i·19-s + (−1.77 − 4.22i)21-s + 6i·23-s + 0.999·25-s + (3.67 + 3.67i)27-s + 6i·29-s + (−2.44 + 5.99i)35-s + 2·37-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + 1.09·5-s + (−0.377 + 0.925i)7-s − 0.999i·9-s + 0.679i·13-s + (−0.774 + 0.774i)15-s + 1.18·17-s − 0.561i·19-s + (−0.387 − 0.921i)21-s + 1.25i·23-s + 0.199·25-s + (0.707 + 0.707i)27-s + 1.11i·29-s + (−0.414 + 1.01i)35-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.363248414\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.363248414\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 12.2iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−9.832326089393160521000969777751, −9.285290311999689178735756141806, −8.611621039020831396549790210836, −7.16615794489163019611965683547, −6.35644266529429407330842607946, −5.50825788420763745272891317105, −5.22651224072673337728531477348, −3.84392691859893675269185208098, −2.81233117704625371871734739836, −1.48102845134630676094188541614,
0.63951071247231076208939433919, 1.78850203735628051610877710490, 3.00212806464306532900267735890, 4.34475638788794329020820104505, 5.46636012062636927595826995859, 6.02753901704502443368209942138, 6.77130199034238236588927035027, 7.65124702148908230339931010677, 8.336393688926796395655646213354, 9.711548202965156038474394865543