| L(s) = 1 | − 2-s − 0.618·3-s + 4-s + 0.618·6-s + 4.85·7-s − 8-s − 2.61·9-s − 3.38·11-s − 0.618·12-s + 0.381·13-s − 4.85·14-s + 16-s − 5.85·17-s + 2.61·18-s − 6.85·19-s − 3.00·21-s + 3.38·22-s + 23-s + 0.618·24-s − 0.381·26-s + 3.47·27-s + 4.85·28-s + 3.70·29-s − 8.85·31-s − 32-s + 2.09·33-s + 5.85·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.252·6-s + 1.83·7-s − 0.353·8-s − 0.872·9-s − 1.01·11-s − 0.178·12-s + 0.105·13-s − 1.29·14-s + 0.250·16-s − 1.41·17-s + 0.617·18-s − 1.57·19-s − 0.654·21-s + 0.721·22-s + 0.208·23-s + 0.126·24-s − 0.0749·26-s + 0.668·27-s + 0.917·28-s + 0.688·29-s − 1.59·31-s − 0.176·32-s + 0.363·33-s + 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 0.381T + 13T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 3.38T + 41T^{2} \) |
| 43 | \( 1 - 6.76T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 - 2.61T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 + 9.70T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−9.168786188905627227989834378990, −8.398898139388876514009909918005, −8.090618644981143833435556887718, −7.03097665815656934479083627691, −6.02491824407672582751981454041, −5.11874925961327733329315326042, −4.36612558428630747289007010596, −2.62801579316177530868689775279, −1.76221570872595072155969042059, 0,
1.76221570872595072155969042059, 2.62801579316177530868689775279, 4.36612558428630747289007010596, 5.11874925961327733329315326042, 6.02491824407672582751981454041, 7.03097665815656934479083627691, 8.090618644981143833435556887718, 8.398898139388876514009909918005, 9.168786188905627227989834378990
