| L(s) = 1 | − 2-s + 1.82·3-s + 4-s − 3.88·5-s − 1.82·6-s − 1.09·7-s − 8-s + 0.315·9-s + 3.88·10-s − 11-s + 1.82·12-s + 3.26·13-s + 1.09·14-s − 7.07·15-s + 16-s + 3.47·17-s − 0.315·18-s − 0.547·19-s − 3.88·20-s − 1.98·21-s + 22-s + 3.83·23-s − 1.82·24-s + 10.1·25-s − 3.26·26-s − 4.88·27-s − 1.09·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.05·3-s + 0.5·4-s − 1.73·5-s − 0.743·6-s − 0.412·7-s − 0.353·8-s + 0.105·9-s + 1.22·10-s − 0.301·11-s + 0.525·12-s + 0.905·13-s + 0.291·14-s − 1.82·15-s + 0.250·16-s + 0.842·17-s − 0.0744·18-s − 0.125·19-s − 0.869·20-s − 0.433·21-s + 0.213·22-s + 0.800·23-s − 0.371·24-s + 2.02·25-s − 0.640·26-s − 0.940·27-s − 0.206·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 + 3.88T + 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 0.547T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 - 7.20T + 29T^{2} \) |
| 31 | \( 1 + 9.62T + 31T^{2} \) |
| 37 | \( 1 - 3.31T + 37T^{2} \) |
| 41 | \( 1 + 0.584T + 41T^{2} \) |
| 43 | \( 1 + 6.84T + 43T^{2} \) |
| 47 | \( 1 - 9.22T + 47T^{2} \) |
| 53 | \( 1 + 9.00T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 9.09T + 61T^{2} \) |
| 67 | \( 1 + 7.44T + 67T^{2} \) |
| 71 | \( 1 - 4.05T + 71T^{2} \) |
| 73 | \( 1 + 1.66T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 7.20T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.133228476388461922281737864755, −7.55592823078997936049620601654, −6.96413454896081348298834484310, −5.99764760260625991040293468630, −4.87966464524127183585195203426, −3.76832739724147374463153838851, −3.37113883005498995558458526539, −2.66177912007590006329903299548, −1.23651045809136327520039552050, 0,
1.23651045809136327520039552050, 2.66177912007590006329903299548, 3.37113883005498995558458526539, 3.76832739724147374463153838851, 4.87966464524127183585195203426, 5.99764760260625991040293468630, 6.96413454896081348298834484310, 7.55592823078997936049620601654, 8.133228476388461922281737864755
