L(s) = 1 | + (0.285 − 0.495i)2-s + (−0.5 + 0.866i)3-s + (0.836 + 1.44i)4-s + (1.33 − 2.31i)5-s + (0.285 + 0.495i)6-s − 3.67·7-s + 2.10·8-s + (−0.499 − 0.866i)9-s + (−0.764 − 1.32i)10-s − 3.81·11-s − 1.67·12-s + (−0.0719 − 0.124i)13-s + (−1.05 + 1.81i)14-s + (1.33 + 2.31i)15-s + (−1.07 + 1.85i)16-s + ⋯ |
L(s) = 1 | + (0.202 − 0.350i)2-s + (−0.288 + 0.499i)3-s + (0.418 + 0.724i)4-s + (0.597 − 1.03i)5-s + (0.116 + 0.202i)6-s − 1.38·7-s + 0.742·8-s + (−0.166 − 0.288i)9-s + (−0.241 − 0.418i)10-s − 1.15·11-s − 0.482·12-s + (−0.0199 − 0.0345i)13-s + (−0.280 + 0.486i)14-s + (0.345 + 0.597i)15-s + (−0.267 + 0.464i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.918414 - 0.00754848i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.918414 - 0.00754848i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-4.24 - 0.990i)T \) |
good | 2 | \( 1 + (-0.285 + 0.495i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.33 + 2.31i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 + (0.0719 + 0.124i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (3.76 + 6.52i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.67 - 4.62i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8.81T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (2.67 - 4.62i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.40 - 2.43i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.00 + 6.94i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.90 - 3.30i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.74 + 9.95i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.69 + 4.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.81 + 11.8i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.172 + 0.299i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.26 - 5.65i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.28T + 83T^{2} \) |
| 89 | \( 1 + (-4.33 - 7.51i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.95 + 5.12i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−15.79132543079098679276192599947, −13.70781247855639843511033799717, −12.79734605495329255649767817502, −12.16506371133357347652857568166, −10.53892121203741385623123343821, −9.600768409773439544157499173348, −8.178099061397265020894718182557, −6.38751472088923433339225279724, −4.83128594404653082423649151182, −3.02711798323333465484932043094,
2.70818058035363871303248431601, 5.60571460097962418484689675051, 6.47121470790871871560557699886, 7.46105943233515286424869484117, 9.850478153649958921899851704353, 10.41478192402783261517544584765, 11.82677434114662715527895155892, 13.43961616221065116784361556092, 13.88484499280457131580449180339, 15.45886773406256492697656751291