L(s) = 1 | + (−2.13 + 1.22i)3-s + (−1.28 − 0.742i)5-s + (−0.129 − 2.64i)7-s + (1.52 − 2.64i)9-s + (4.37 − 2.52i)11-s − 2.58i·13-s + 3.65·15-s + (−0.629 − 1.09i)17-s + (−2.68 − 1.54i)19-s + (3.52 + 5.47i)21-s + (−0.697 + 1.20i)23-s + (−1.39 − 2.41i)25-s + 0.126i·27-s + 0.638i·29-s + (−1.82 − 3.16i)31-s + ⋯ |
L(s) = 1 | + (−1.22 + 0.710i)3-s + (−0.575 − 0.332i)5-s + (−0.0490 − 0.998i)7-s + (0.508 − 0.880i)9-s + (1.31 − 0.760i)11-s − 0.717i·13-s + 0.943·15-s + (−0.152 − 0.264i)17-s + (−0.615 − 0.355i)19-s + (0.769 + 1.19i)21-s + (−0.145 + 0.252i)23-s + (−0.279 − 0.483i)25-s + 0.0243i·27-s + 0.118i·29-s + (−0.328 − 0.568i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.486781 - 0.364822i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.486781 - 0.364822i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.129 + 2.64i)T \) |
good | 3 | \( 1 + (2.13 - 1.22i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.28 + 0.742i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.37 + 2.52i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.58iT - 13T^{2} \) |
| 17 | \( 1 + (0.629 + 1.09i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.68 + 1.54i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.697 - 1.20i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.638iT - 29T^{2} \) |
| 31 | \( 1 + (1.82 + 3.16i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.21 + 3.01i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 1.02iT - 43T^{2} \) |
| 47 | \( 1 + (5.48 - 9.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.99 + 2.88i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.01 + 1.74i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.1 - 6.44i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.443 - 0.256i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.41T + 71T^{2} \) |
| 73 | \( 1 + (4.94 + 8.56i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.35 + 7.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.97iT - 83T^{2} \) |
| 89 | \( 1 + (-1.29 + 2.23i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 1.57T + 97T^{2} \) |
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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
−11.74707624304789255728967070121, −11.14698719339297917894662945584, −10.38562068481014179250448950452, −9.313644209733722783598992069621, −8.068691270275096510294267537417, −6.77529864523697320885807796833, −5.78404146536148031188477763220, −4.51561503782930988384248410072, −3.74847564408547277642612560083, −0.60843294209001267401494823743,
1.82471537950692023762108939149, 3.95939416872215830529147323441, 5.36402017015951605085208621590, 6.50390441287699946004958278039, 7.00234316929147723983249008319, 8.462350948291587324703643849915, 9.550463300984065431362168004137, 10.90292037646279887918416959655, 11.81090787632649219401668667663, 12.07296322074608178403398928780