L(s) = 1 | + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + 3·8-s + (2 + 3.46i)11-s + (0.500 − 0.866i)16-s + 3.99·22-s + (4 − 6.92i)23-s + (2.5 + 4.33i)25-s − 2·29-s + (2.50 + 4.33i)32-s + (3 − 5.19i)37-s − 12·43-s + (−1.99 + 3.46i)44-s + (−3.99 − 6.92i)46-s + 5·50-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + 1.06·8-s + (0.603 + 1.04i)11-s + (0.125 − 0.216i)16-s + 0.852·22-s + (0.834 − 1.44i)23-s + (0.5 + 0.866i)25-s − 0.371·29-s + (0.441 + 0.765i)32-s + (0.493 − 0.854i)37-s − 1.82·43-s + (−0.301 + 0.522i)44-s + (−0.589 − 1.02i)46-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.92943 - 0.122442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.92943 - 0.122442i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5 + 8.66i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.22118699307712351797105705482, −10.42260184288219637129287499863, −9.423907224831202791519044739523, −8.394894458611223772036190976834, −7.29139196053559166721759899932, −6.62665534373549917236332618591, −5.01970630315614486339063133602, −4.13031728033590283530223709279, −2.97309757549757489362482940577, −1.72308377574922347240824843108,
1.36244013126385186572929554582, 3.17051580992645947470498223898, 4.53634707812790858538234661528, 5.58751412495102888234703243537, 6.36579893512035772327223491545, 7.24377531406218404291946005590, 8.287968396041543380977247870036, 9.308040957925004218281811246323, 10.30471708388913224954778517263, 11.18803891444180594550764081904