L(s) = 1 | + (−0.864 − 2.87i)3-s − 9.02i·7-s + (−7.50 + 4.96i)9-s − 21.8i·11-s − 21.6i·13-s + 12.1·17-s + 3.03·19-s + (−25.9 + 7.80i)21-s + 28.5·23-s + (20.7 + 17.2i)27-s − 12.0i·29-s − 2.19·31-s + (−62.8 + 18.9i)33-s + 0.839i·37-s + (−62.2 + 18.7i)39-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.957i)3-s − 1.28i·7-s + (−0.833 + 0.551i)9-s − 1.98i·11-s − 1.66i·13-s + 0.712·17-s + 0.159·19-s + (−1.23 + 0.371i)21-s + 1.24·23-s + (0.768 + 0.639i)27-s − 0.415i·29-s − 0.0706·31-s + (−1.90 + 0.573i)33-s + 0.0227i·37-s + (−1.59 + 0.480i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.537044304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537044304\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.864 + 2.87i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 9.02iT - 49T^{2} \) |
| 11 | \( 1 + 21.8iT - 121T^{2} \) |
| 13 | \( 1 + 21.6iT - 169T^{2} \) |
| 17 | \( 1 - 12.1T + 289T^{2} \) |
| 19 | \( 1 - 3.03T + 361T^{2} \) |
| 23 | \( 1 - 28.5T + 529T^{2} \) |
| 29 | \( 1 + 12.0iT - 841T^{2} \) |
| 31 | \( 1 + 2.19T + 961T^{2} \) |
| 37 | \( 1 - 0.839iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 35.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 12.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 22.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 9.13T + 2.80e3T^{2} \) |
| 59 | \( 1 + 80.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 52.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 7.46T + 6.24e3T^{2} \) |
| 83 | \( 1 - 82.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 27.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 114. iT - 9.40e3T^{2} \) |
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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
−8.911979746089779510312948479133, −7.990079230258679940445043357397, −7.66763604423839516589228478658, −6.61963483296964143348688283016, −5.82072782098981670916241117707, −5.10844716357545026149330300911, −3.53030280788958131418406945875, −2.92057093700100027245135455959, −1.02747248006787136289273415214, −0.58279790206938168146175737074,
1.77867149920851320737202319909, 2.82393071757805776425783447547, 4.13436106084374609279143498346, 4.84224645894727437360635317747, 5.57484498674159226812254384960, 6.63262017017730058866190207196, 7.40734931470455617202183829883, 8.790708806184720049766542960835, 9.228920564015432474668482883291, 9.800427732420380005461951907595