| L(s) = 1 | + 2-s − 1.84·3-s + 4-s − 5-s − 1.84·6-s − 5.01·7-s + 8-s + 0.414·9-s − 10-s − 4.69·11-s − 1.84·12-s + 3.83·13-s − 5.01·14-s + 1.84·15-s + 16-s − 5.78·17-s + 0.414·18-s + 5.19·19-s − 20-s + 9.25·21-s − 4.69·22-s + 6.40·23-s − 1.84·24-s + 25-s + 3.83·26-s + 4.77·27-s − 5.01·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.06·3-s + 0.5·4-s − 0.447·5-s − 0.754·6-s − 1.89·7-s + 0.353·8-s + 0.138·9-s − 0.316·10-s − 1.41·11-s − 0.533·12-s + 1.06·13-s − 1.33·14-s + 0.477·15-s + 0.250·16-s − 1.40·17-s + 0.0976·18-s + 1.19·19-s − 0.223·20-s + 2.02·21-s − 1.00·22-s + 1.33·23-s − 0.377·24-s + 0.200·25-s + 0.752·26-s + 0.919·27-s − 0.946·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 1.84T + 3T^{2} \) |
| 7 | \( 1 + 5.01T + 7T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 - 3.83T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 - 6.40T + 23T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 + 7.74T + 41T^{2} \) |
| 43 | \( 1 - 0.904T + 43T^{2} \) |
| 47 | \( 1 - 0.257T + 47T^{2} \) |
| 53 | \( 1 + 3.83T + 53T^{2} \) |
| 59 | \( 1 - 4.16T + 59T^{2} \) |
| 61 | \( 1 + 2.99T + 61T^{2} \) |
| 67 | \( 1 + 1.92T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 8.36T + 97T^{2} \) |
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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
−6.94537774491825474321855022161, −6.53398540560926640410736908268, −6.04360775327742847174187576290, −5.24068476672548678858500091700, −4.81688824701845601627966015169, −3.75064038521084321737536799683, −3.10447262354289391171114300363, −2.58824464373040444120139441430, −0.903409693238353169384284592919, 0,
0.903409693238353169384284592919, 2.58824464373040444120139441430, 3.10447262354289391171114300363, 3.75064038521084321737536799683, 4.81688824701845601627966015169, 5.24068476672548678858500091700, 6.04360775327742847174187576290, 6.53398540560926640410736908268, 6.94537774491825474321855022161
