L(s) = 1 | + (−0.619 − 2.93i)3-s + (−4.48 + 2.21i)5-s − 4.63i·7-s + (−8.23 + 3.63i)9-s + 15.0·11-s − 3.35·13-s + (9.27 + 11.7i)15-s − 11.1·17-s + 22.7i·19-s + (−13.5 + 2.87i)21-s − 36.1·23-s + (15.2 − 19.8i)25-s + (15.7 + 21.9i)27-s − 21.5·29-s − 28.3·31-s + ⋯ |
L(s) = 1 | + (−0.206 − 0.978i)3-s + (−0.896 + 0.442i)5-s − 0.661i·7-s + (−0.914 + 0.404i)9-s + 1.37·11-s − 0.258·13-s + (0.618 + 0.785i)15-s − 0.655·17-s + 1.19i·19-s + (−0.647 + 0.136i)21-s − 1.57·23-s + (0.608 − 0.793i)25-s + (0.584 + 0.811i)27-s − 0.743·29-s − 0.915·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.044551600\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.044551600\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.619 + 2.93i)T \) |
| 5 | \( 1 + (4.48 - 2.21i)T \) |
good | 7 | \( 1 + 4.63iT - 49T^{2} \) |
| 11 | \( 1 - 15.0T + 121T^{2} \) |
| 13 | \( 1 + 3.35T + 169T^{2} \) |
| 17 | \( 1 + 11.1T + 289T^{2} \) |
| 19 | \( 1 - 22.7iT - 361T^{2} \) |
| 23 | \( 1 + 36.1T + 529T^{2} \) |
| 29 | \( 1 + 21.5T + 841T^{2} \) |
| 31 | \( 1 + 28.3T + 961T^{2} \) |
| 37 | \( 1 - 69.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 36.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 29.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 8.07T + 3.48e3T^{2} \) |
| 61 | \( 1 + 10.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 91.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 79.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 86.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 10.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 139. iT - 9.40e3T^{2} \) |
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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
−9.933244808651583471114657841291, −8.910116921869795527966177885938, −7.84558961048025381627258498771, −7.50041486191533067828613997252, −6.51489685370403843345184545029, −5.93602592153399143871056053340, −4.31017893474308420946078961143, −3.68442547393839018116665170006, −2.22970375617442390137292391224, −0.948005682463907380279249205294,
0.43986140279755618563149873121, 2.37001428179130389209017534004, 3.82958878438732701806712882176, 4.23588187594082758290348000767, 5.31309855616597616226939961216, 6.19312684421658135062725083047, 7.27248268542210609185320991255, 8.355814377262877710801145149902, 9.229507051244111946770043080587, 9.366026237860151354465712191643