L(s) = 1 | + (0.358 − 0.207i)2-s + (−0.914 + 1.58i)4-s + (−1.46 − 2.53i)5-s + 1.58i·8-s + (−1.05 − 0.606i)10-s + (−4.18 − 2.41i)11-s − 2.93i·13-s + (−1.49 − 2.59i)16-s + (3.53 − 6.12i)17-s + (−5.07 + 2.93i)19-s + 5.35·20-s − 2·22-s + (−1.73 + i)23-s + (−1.79 + 3.10i)25-s + (−0.606 − 1.05i)26-s + ⋯ |
L(s) = 1 | + (0.253 − 0.146i)2-s + (−0.457 + 0.791i)4-s + (−0.655 − 1.13i)5-s + 0.560i·8-s + (−0.332 − 0.191i)10-s + (−1.26 − 0.727i)11-s − 0.812i·13-s + (−0.374 − 0.649i)16-s + (0.857 − 1.48i)17-s + (−1.16 + 0.672i)19-s + 1.19·20-s − 0.426·22-s + (−0.361 + 0.208i)23-s + (−0.358 + 0.621i)25-s + (−0.119 − 0.206i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.256163 - 0.588780i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.256163 - 0.588780i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.358 + 0.207i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.46 + 2.53i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.18 + 2.41i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 + (-3.53 + 6.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.07 - 2.93i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.828iT - 29T^{2} \) |
| 31 | \( 1 + (5.07 + 2.93i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.70 - 4.68i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 + (2.93 + 5.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.12 - 3.53i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.93 - 5.07i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.05 - 0.606i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.24 + 7.34i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.828iT - 71T^{2} \) |
| 73 | \( 1 + (-6.12 - 3.53i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.828 + 1.43i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + (-5.60 - 9.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7.07iT - 97T^{2} \) |
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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
−10.97204606817643885708878929504, −9.821279382255632269069987750292, −8.716823305134050776949973333569, −8.075783985321762271901011815324, −7.54252141618576649915626659247, −5.63714936489767859610861949279, −4.93048906883958318450694502738, −3.86768052742692651584398732185, −2.76992830710026646516335185634, −0.36242379795135877698593788068,
2.13005090385928615795698129694, 3.68718387199142075207688638561, 4.64101875823616561746054838315, 5.84355533130047615674993550995, 6.76274414905622134782716832956, 7.63232525806555307111840068791, 8.715555681452743603142558622389, 9.920981798370689165443630862981, 10.59919211949422075615189379607, 11.12500462939317025790056237468