| L(s) = 1 | + 2-s − 3.30·3-s + 4-s − 3.30·6-s + 2.60·7-s + 8-s + 7.90·9-s − 2.30·11-s − 3.30·12-s − 1.30·13-s + 2.60·14-s + 16-s + 6·17-s + 7.90·18-s + 2·19-s − 8.60·21-s − 2.30·22-s − 3.90·23-s − 3.30·24-s − 1.30·26-s − 16.2·27-s + 2.60·28-s − 3.90·29-s − 0.302·31-s + 32-s + 7.60·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.90·3-s + 0.5·4-s − 1.34·6-s + 0.984·7-s + 0.353·8-s + 2.63·9-s − 0.694·11-s − 0.953·12-s − 0.361·13-s + 0.696·14-s + 0.250·16-s + 1.45·17-s + 1.86·18-s + 0.458·19-s − 1.87·21-s − 0.490·22-s − 0.814·23-s − 0.674·24-s − 0.255·26-s − 3.11·27-s + 0.492·28-s − 0.725·29-s − 0.0543·31-s + 0.176·32-s + 1.32·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.594412425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.594412425\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.90T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + 0.302T + 31T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 + 0.605T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 - 3.51T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−9.718053053128716410831979470838, −8.017264901818200086858293348505, −7.50572484000297588098484172036, −6.65848133781939143594087281410, −5.53317025922360171796568306061, −5.47882644265647658275526629298, −4.62675218985074459374435324413, −3.71827902306834164516962095765, −2.06485302664892115251413844573, −0.878006943511392673067401385117,
0.878006943511392673067401385117, 2.06485302664892115251413844573, 3.71827902306834164516962095765, 4.62675218985074459374435324413, 5.47882644265647658275526629298, 5.53317025922360171796568306061, 6.65848133781939143594087281410, 7.50572484000297588098484172036, 8.017264901818200086858293348505, 9.718053053128716410831979470838
