L(s) = 1 | + (0.5 − 0.866i)3-s + (−1.5 + 2.59i)5-s + (1 + 1.73i)9-s + 4·11-s + (2.5 + 4.33i)13-s + (1.5 + 2.59i)15-s + (2.5 − 4.33i)17-s + (−4 + 1.73i)19-s + (−0.5 − 0.866i)23-s + (−2 − 3.46i)25-s + 5·27-s + (−1.5 − 2.59i)29-s − 4·31-s + (2 − 3.46i)33-s + 2·37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.670 + 1.16i)5-s + (0.333 + 0.577i)9-s + 1.20·11-s + (0.693 + 1.20i)13-s + (0.387 + 0.670i)15-s + (0.606 − 1.05i)17-s + (−0.917 + 0.397i)19-s + (−0.104 − 0.180i)23-s + (−0.400 − 0.692i)25-s + 0.962·27-s + (−0.278 − 0.482i)29-s − 0.718·31-s + (0.348 − 0.603i)33-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30100 + 0.417818i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30100 + 0.417818i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.5 + 11.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.5 + 14.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.58697672930021961809791810809, −11.15891910130723296563063500809, −9.986088556969341745235437774794, −8.895922924224607761627142245965, −7.79438621243966777603425850245, −6.98457395880764282566472946613, −6.29734339844758105692908509347, −4.43120971616653911414314481139, −3.38126928343609139256602274873, −1.87332807902193277179806298658,
1.15031396200215619900310020609, 3.58997165969822650202113996921, 4.18010988530899099844113085465, 5.50674024609109312674876864737, 6.72668074657122062497307922331, 8.194093678871529073832657475720, 8.692836682834387140070136896912, 9.614369621802696501389115401608, 10.64707559479990751395030112701, 11.73642150993963399831469577158