L(s) = 1 | + 7.67i·5-s − 7.21·7-s − 6.05i·11-s − 2.29·13-s + 21.8i·17-s − 34.8·19-s − 21.5i·23-s − 33.8·25-s − 10.9i·29-s + 37.6·31-s − 55.3i·35-s + 34.8·37-s + 13.3i·41-s − 60.5·43-s − 3.34i·47-s + ⋯ |
L(s) = 1 | + 1.53i·5-s − 1.03·7-s − 0.550i·11-s − 0.176·13-s + 1.28i·17-s − 1.83·19-s − 0.935i·23-s − 1.35·25-s − 0.376i·29-s + 1.21·31-s − 1.58i·35-s + 0.941·37-s + 0.325i·41-s − 1.40·43-s − 0.0712i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6874242122\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6874242122\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 7.67iT - 25T^{2} \) |
| 7 | \( 1 + 7.21T + 49T^{2} \) |
| 11 | \( 1 + 6.05iT - 121T^{2} \) |
| 13 | \( 1 + 2.29T + 169T^{2} \) |
| 17 | \( 1 - 21.8iT - 289T^{2} \) |
| 19 | \( 1 + 34.8T + 361T^{2} \) |
| 23 | \( 1 + 21.5iT - 529T^{2} \) |
| 29 | \( 1 + 10.9iT - 841T^{2} \) |
| 31 | \( 1 - 37.6T + 961T^{2} \) |
| 37 | \( 1 - 34.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 13.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 60.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 3.34iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 37.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 25.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 37.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 77.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 55.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.43iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 52.8T + 9.40e3T^{2} \) |
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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
−8.536088804258059679678842837743, −8.044317341581797440194882717043, −6.84566670218608083777399086157, −6.37259942318614987656873862044, −6.06570971874970174363068736979, −4.51523946146724309543378100182, −3.63025223998863736534178336211, −2.90305705142305113887467483286, −2.08292591629591784892860141045, −0.21045616311403144164334553071,
0.820434853989070251230620953222, 2.04575244351228684606999789139, 3.16809653777916044784268004572, 4.35800145788173591400202492561, 4.80203980764347546047564179482, 5.77750014835355316568960668361, 6.58598151398478708564106306674, 7.41944092874969791899845097061, 8.355498045006431755685252270505, 8.962735330035581497868680772357