L(s) = 1 | + 1.82·2-s − 2.85·3-s + 1.32·4-s + 5-s − 5.21·6-s + 7-s − 1.23·8-s + 5.16·9-s + 1.82·10-s − 0.0862·11-s − 3.78·12-s − 0.762·13-s + 1.82·14-s − 2.85·15-s − 4.89·16-s − 0.690·17-s + 9.42·18-s + 0.149·19-s + 1.32·20-s − 2.85·21-s − 0.157·22-s − 0.700·23-s + 3.52·24-s + 25-s − 1.39·26-s − 6.20·27-s + 1.32·28-s + ⋯ |
L(s) = 1 | + 1.28·2-s − 1.65·3-s + 0.662·4-s + 0.447·5-s − 2.12·6-s + 0.377·7-s − 0.435·8-s + 1.72·9-s + 0.576·10-s − 0.0260·11-s − 1.09·12-s − 0.211·13-s + 0.487·14-s − 0.738·15-s − 1.22·16-s − 0.167·17-s + 2.22·18-s + 0.0343·19-s + 0.296·20-s − 0.623·21-s − 0.0335·22-s − 0.146·23-s + 0.718·24-s + 0.200·25-s − 0.272·26-s − 1.19·27-s + 0.250·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 3 | \( 1 + 2.85T + 3T^{2} \) |
| 11 | \( 1 + 0.0862T + 11T^{2} \) |
| 13 | \( 1 + 0.762T + 13T^{2} \) |
| 17 | \( 1 + 0.690T + 17T^{2} \) |
| 19 | \( 1 - 0.149T + 19T^{2} \) |
| 23 | \( 1 + 0.700T + 23T^{2} \) |
| 29 | \( 1 - 0.886T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 6.12T + 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 + 7.84T + 53T^{2} \) |
| 59 | \( 1 + 8.01T + 59T^{2} \) |
| 61 | \( 1 + 0.455T + 61T^{2} \) |
| 67 | \( 1 + 8.65T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 7.74T + 73T^{2} \) |
| 79 | \( 1 + 3.32T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 8.87T + 89T^{2} \) |
| 97 | \( 1 - 3.04T + 97T^{2} \) |
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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
−7.04845143078760868608327697713, −6.43178991301319939838987455016, −5.88589571166975740981974477107, −5.44040575888269343960501581750, −4.62462607351225883543814267217, −4.45239481583231202331360876829, −3.32019417359286756399686498036, −2.34141295111498945287706158741, −1.23349368043112522581354222494, 0,
1.23349368043112522581354222494, 2.34141295111498945287706158741, 3.32019417359286756399686498036, 4.45239481583231202331360876829, 4.62462607351225883543814267217, 5.44040575888269343960501581750, 5.88589571166975740981974477107, 6.43178991301319939838987455016, 7.04845143078760868608327697713