L(s) = 1 | − 2.77i·2-s − 5.71·4-s − 2.71i·7-s + 10.3i·8-s + 2.71·11-s − i·13-s − 7.55·14-s + 17.2·16-s + 2.83i·17-s + 3.55·19-s − 7.55i·22-s − 4.83i·23-s − 2.77·26-s + 15.5i·28-s + 6·29-s + ⋯ |
L(s) = 1 | − 1.96i·2-s − 2.85·4-s − 1.02i·7-s + 3.65i·8-s + 0.820·11-s − 0.277i·13-s − 2.01·14-s + 4.31·16-s + 0.688i·17-s + 0.816·19-s − 1.61i·22-s − 1.00i·23-s − 0.544·26-s + 2.93i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.493919926\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.493919926\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + 2.77iT - 2T^{2} \) |
| 7 | \( 1 + 2.71iT - 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 17 | \( 1 - 2.83iT - 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 + 4.83iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 4.27iT - 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 11.5iT - 47T^{2} \) |
| 53 | \( 1 - 1.16iT - 53T^{2} \) |
| 59 | \( 1 + 2.11T + 59T^{2} \) |
| 61 | \( 1 - 6.60T + 61T^{2} \) |
| 67 | \( 1 - 1.88iT - 67T^{2} \) |
| 71 | \( 1 - 6.71T + 71T^{2} \) |
| 73 | \( 1 + 9.11iT - 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.11iT - 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 + 10.8iT - 97T^{2} \) |
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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
−8.573099513252450392106436405455, −7.974933286765617223893510788650, −6.84991444466345884937329693615, −5.75255170585181534845119633154, −4.67059065759789347594728648673, −4.15871186650619138345966010974, −3.41103778337391791688884707364, −2.54864403268418432952637271477, −1.37767907091358237423272037090, −0.63901267862910274700415173595,
1.09969763973424146601150858720, 2.98667997836064838875416576919, 4.03448158768539227749595663499, 4.95909224279865526712281394026, 5.43101118212270929531921955803, 6.36346368250773820226402589473, 6.74334827128507410516052002276, 7.62466547674301124203068261264, 8.364867982997215590950798186665, 8.870145687593956449801058605638