L(s) = 1 | − 0.485·2-s − 0.214·3-s − 1.76·4-s − 5-s + 0.104·6-s + 3.34·7-s + 1.82·8-s − 2.95·9-s + 0.485·10-s + 11-s + 0.378·12-s + 5.34·13-s − 1.62·14-s + 0.214·15-s + 2.64·16-s − 7.63·17-s + 1.43·18-s − 5.15·19-s + 1.76·20-s − 0.718·21-s − 0.485·22-s − 1.25·23-s − 0.391·24-s + 25-s − 2.59·26-s + 1.27·27-s − 5.90·28-s + ⋯ |
L(s) = 1 | − 0.342·2-s − 0.123·3-s − 0.882·4-s − 0.447·5-s + 0.0425·6-s + 1.26·7-s + 0.645·8-s − 0.984·9-s + 0.153·10-s + 0.301·11-s + 0.109·12-s + 1.48·13-s − 0.433·14-s + 0.0554·15-s + 0.660·16-s − 1.85·17-s + 0.337·18-s − 1.18·19-s + 0.394·20-s − 0.156·21-s − 0.103·22-s − 0.262·23-s − 0.0800·24-s + 0.200·25-s − 0.508·26-s + 0.245·27-s − 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 0.485T + 2T^{2} \) |
| 3 | \( 1 + 0.214T + 3T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 + 7.63T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 1.02T + 43T^{2} \) |
| 47 | \( 1 - 9.09T + 47T^{2} \) |
| 53 | \( 1 + 0.0737T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 0.307T + 61T^{2} \) |
| 67 | \( 1 - 2.23T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 79 | \( 1 + 1.94T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 1.89T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.276204645749824187080088323892, −7.71181434890175160702248098418, −6.47154884985260735642145596453, −5.95180650827622673882743828271, −4.80408279579655310030434300358, −4.41482172221479907201951914987, −3.64510762178238129186886295556, −2.30824093867538827419414544938, −1.23039743491141504106584841249, 0,
1.23039743491141504106584841249, 2.30824093867538827419414544938, 3.64510762178238129186886295556, 4.41482172221479907201951914987, 4.80408279579655310030434300358, 5.95180650827622673882743828271, 6.47154884985260735642145596453, 7.71181434890175160702248098418, 8.276204645749824187080088323892