| L(s) = 1 | + (0.142 + 0.989i)3-s + (−0.991 − 0.291i)5-s + (−2.31 + 2.66i)7-s + (−0.959 + 0.281i)9-s + (−1.40 + 0.899i)11-s + (−3.21 − 3.71i)13-s + (0.147 − 1.02i)15-s + (0.654 − 1.43i)17-s + (−3.15 − 6.89i)19-s + (−2.96 − 1.90i)21-s + (1.47 + 4.56i)23-s + (−3.30 − 2.12i)25-s + (−0.415 − 0.909i)27-s + (−3.54 + 7.75i)29-s + (−0.990 + 6.88i)31-s + ⋯ |
| L(s) = 1 | + (0.0821 + 0.571i)3-s + (−0.443 − 0.130i)5-s + (−0.873 + 1.00i)7-s + (−0.319 + 0.0939i)9-s + (−0.422 + 0.271i)11-s + (−0.891 − 1.02i)13-s + (0.0379 − 0.264i)15-s + (0.158 − 0.347i)17-s + (−0.722 − 1.58i)19-s + (−0.647 − 0.416i)21-s + (0.307 + 0.951i)23-s + (−0.661 − 0.425i)25-s + (−0.0799 − 0.175i)27-s + (−0.657 + 1.43i)29-s + (−0.177 + 1.23i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0241383 - 0.229229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0241383 - 0.229229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-1.47 - 4.56i)T \) |
| good | 5 | \( 1 + (0.991 + 0.291i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (2.31 - 2.66i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (1.40 - 0.899i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.21 + 3.71i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.654 + 1.43i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (3.15 + 6.89i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.54 - 7.75i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.990 - 6.88i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-8.02 + 2.35i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (11.4 + 3.37i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.212 - 1.47i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 3.84T + 47T^{2} \) |
| 53 | \( 1 + (4.28 - 4.94i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-8.34 - 9.63i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.641 + 4.45i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.57 + 2.29i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (2.75 + 1.76i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (4.00 + 8.76i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-11.3 - 13.0i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.51 + 0.737i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.21 - 15.4i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (4.37 + 1.28i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.18506060947949267731666829293, −10.29799818825286820682978842425, −9.429624918580359353888674305664, −8.826148671857953948594202190301, −7.73991667951205608692443851827, −6.77223053831126568064780317251, −5.47951255593132008016524234035, −4.86540065212504263752549594160, −3.37501270205553572779059799617, −2.56311129094603802838678389844,
0.12121962829091002820052917602, 2.07274435300861277619103675819, 3.53337488888847052858772208034, 4.37164811750492523008107824987, 5.97801244348845209939249638671, 6.72364755535860474046237755919, 7.63812035206603367236147222008, 8.261373193059270390025473964743, 9.674961354953254610509114046677, 10.15319197748440514415010372277
