L(s) = 1 | − 2.62·2-s − 1.27·3-s + 4.88·4-s + 1.41·5-s + 3.33·6-s + 7-s − 7.56·8-s − 1.38·9-s − 3.72·10-s + 1.00·11-s − 6.20·12-s + 2.84·13-s − 2.62·14-s − 1.80·15-s + 10.0·16-s − 0.358·17-s + 3.62·18-s + 5.52·19-s + 6.92·20-s − 1.27·21-s − 2.63·22-s − 1.69·23-s + 9.61·24-s − 2.98·25-s − 7.45·26-s + 5.57·27-s + 4.88·28-s + ⋯ |
L(s) = 1 | − 1.85·2-s − 0.734·3-s + 2.44·4-s + 0.634·5-s + 1.36·6-s + 0.377·7-s − 2.67·8-s − 0.461·9-s − 1.17·10-s + 0.302·11-s − 1.79·12-s + 0.787·13-s − 0.701·14-s − 0.465·15-s + 2.52·16-s − 0.0869·17-s + 0.855·18-s + 1.26·19-s + 1.54·20-s − 0.277·21-s − 0.561·22-s − 0.354·23-s + 1.96·24-s − 0.597·25-s − 1.46·26-s + 1.07·27-s + 0.922·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7961432041\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7961432041\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 - 1.00T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 + 0.358T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 - 5.40T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 - 7.89T + 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 + 3.18T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 0.388T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 + 8.67T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 - 8.98T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−8.025412655777920745716837099089, −7.73328716148518438160279030140, −6.58682657625689480394642170505, −6.23468689835074086452548199202, −5.63807003156939551103433552369, −4.62469992439406741562036303397, −3.22655124562232747460184566705, −2.39672023488953165143105256122, −1.38093201096877405110550289899, −0.71524365555508019206886531501,
0.71524365555508019206886531501, 1.38093201096877405110550289899, 2.39672023488953165143105256122, 3.22655124562232747460184566705, 4.62469992439406741562036303397, 5.63807003156939551103433552369, 6.23468689835074086452548199202, 6.58682657625689480394642170505, 7.73328716148518438160279030140, 8.025412655777920745716837099089