L(s) = 1 | + (−0.315 − 0.182i)3-s + (−1.58 + 2.11i)7-s + (−1.43 − 2.48i)9-s + (−0.682 + 1.18i)11-s + 2.63i·13-s + (−1.93 − 1.11i)17-s + (−3.11 − 5.39i)19-s + (0.886 − 0.378i)21-s + (−5.71 + 3.29i)23-s + 2.13i·27-s − 5.50·29-s + (−2.25 + 3.89i)31-s + (0.430 − 0.248i)33-s + (−1.81 + 1.04i)37-s + (0.480 − 0.831i)39-s + ⋯ |
L(s) = 1 | + (−0.182 − 0.105i)3-s + (−0.600 + 0.799i)7-s + (−0.477 − 0.827i)9-s + (−0.205 + 0.356i)11-s + 0.730i·13-s + (−0.468 − 0.270i)17-s + (−0.714 − 1.23i)19-s + (0.193 − 0.0825i)21-s + (−1.19 + 0.687i)23-s + 0.411i·27-s − 1.02·29-s + (−0.404 + 0.700i)31-s + (0.0749 − 0.0432i)33-s + (−0.298 + 0.172i)37-s + (0.0769 − 0.133i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0217411 + 0.177522i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0217411 + 0.177522i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.58 - 2.11i)T \) |
good | 3 | \( 1 + (0.315 + 0.182i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (0.682 - 1.18i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.63iT - 13T^{2} \) |
| 17 | \( 1 + (1.93 + 1.11i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.11 + 5.39i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.71 - 3.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 + (2.25 - 3.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.81 - 1.04i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.32T + 41T^{2} \) |
| 43 | \( 1 - 1.86iT - 43T^{2} \) |
| 47 | \( 1 + (5.94 - 3.43i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.78 + 5.06i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.817 - 1.41i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0197 + 0.0341i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.82 - 3.36i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.27T + 71T^{2} \) |
| 73 | \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.66 - 4.61i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.7iT - 83T^{2} \) |
| 89 | \( 1 + (-0.433 - 0.750i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 16.0iT - 97T^{2} \) |
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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
−11.07893491341415392360759094439, −9.693089055154004211910916548266, −9.252310585305099143938674952144, −8.441502356112807758104655634151, −7.15693566926766723655190542941, −6.41045451659538567606393860013, −5.62012713016835507141202568214, −4.43681595905597864091841474099, −3.22447498408370107113307477000, −2.06450236633736994350945618332,
0.087911403545666118111851434217, 2.12144077633026346225763942704, 3.49085421330557671543741896192, 4.42164430829851612435987968996, 5.70725282292187072764518599280, 6.29742267737982035507134006512, 7.65750944448750227968620982838, 8.109427854158442001028310955872, 9.263046986472416216484969554539, 10.41585960744886553409958114182