L(s) = 1 | + (−0.893 + 0.448i)2-s + (−0.973 + 0.230i)3-s + (0.597 − 0.802i)4-s + (−0.286 + 0.957i)5-s + (0.766 − 0.642i)6-s + (0.543 − 1.26i)7-s + (−0.173 + 0.984i)8-s + (0.893 − 0.448i)9-s + (−0.173 − 0.984i)10-s + (−0.396 + 0.918i)12-s + (0.0798 + 1.37i)14-s + (0.0581 − 0.998i)15-s + (−0.286 − 0.957i)16-s + (−0.597 + 0.802i)18-s + (0.597 + 0.802i)20-s + (−0.238 + 1.35i)21-s + ⋯ |
L(s) = 1 | + (−0.893 + 0.448i)2-s + (−0.973 + 0.230i)3-s + (0.597 − 0.802i)4-s + (−0.286 + 0.957i)5-s + (0.766 − 0.642i)6-s + (0.543 − 1.26i)7-s + (−0.173 + 0.984i)8-s + (0.893 − 0.448i)9-s + (−0.173 − 0.984i)10-s + (−0.396 + 0.918i)12-s + (0.0798 + 1.37i)14-s + (0.0581 − 0.998i)15-s + (−0.286 − 0.957i)16-s + (−0.597 + 0.802i)18-s + (0.597 + 0.802i)20-s + (−0.238 + 1.35i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4876106148\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4876106148\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.893 - 0.448i)T \) |
| 3 | \( 1 + (0.973 - 0.230i)T \) |
| 5 | \( 1 + (0.286 - 0.957i)T \) |
good | 7 | \( 1 + (-0.543 + 1.26i)T + (-0.686 - 0.727i)T^{2} \) |
| 11 | \( 1 + (-0.893 - 0.448i)T^{2} \) |
| 13 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
| 17 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 19 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.313 + 0.727i)T + (-0.686 + 0.727i)T^{2} \) |
| 29 | \( 1 + (-0.115 + 1.98i)T + (-0.993 - 0.116i)T^{2} \) |
| 31 | \( 1 + (-0.973 - 0.230i)T^{2} \) |
| 37 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 41 | \( 1 + (1.49 + 0.749i)T + (0.597 + 0.802i)T^{2} \) |
| 43 | \( 1 + (-0.238 + 0.252i)T + (-0.0581 - 0.998i)T^{2} \) |
| 47 | \( 1 + (-1.77 + 0.207i)T + (0.973 - 0.230i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.893 + 0.448i)T^{2} \) |
| 61 | \( 1 + (-1.16 - 1.56i)T + (-0.286 + 0.957i)T^{2} \) |
| 67 | \( 1 + (0.00676 + 0.116i)T + (-0.993 + 0.116i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (-0.597 + 0.802i)T^{2} \) |
| 83 | \( 1 + (-1.49 + 0.749i)T + (0.597 - 0.802i)T^{2} \) |
| 89 | \( 1 + (-0.310 + 1.76i)T + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.835 - 0.549i)T^{2} \) |
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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
−9.903792135539146533957437962365, −8.681726031814934629710240378291, −7.65890538601697292865095855536, −7.25046887384663453582121956062, −6.48117350406648857933045302386, −5.77932894224018660806013317931, −4.62118440219927889006499576840, −3.81079905210027962299202007440, −2.18130293867593845525090514902, −0.66238698761936109839818916463,
1.23224079054496225741971714446, 2.13346800231475769960074424320, 3.62079909589150927360235903223, 4.87432552356743439709070167407, 5.46633530724101247554583971666, 6.46337817073513459866629229360, 7.43735380221192760333312191213, 8.194825194663170845265341986970, 8.864508155198985451900585575983, 9.520287376920181557746186949629