| L(s) = 1 | + 1.72·2-s + 3-s + 0.982·4-s + 1.72·6-s + 5.06·7-s − 1.75·8-s + 9-s − 5.92·11-s + 0.982·12-s − 6.03·13-s + 8.75·14-s − 4.99·16-s − 4.06·17-s + 1.72·18-s + 2.30·19-s + 5.06·21-s − 10.2·22-s − 3.79·23-s − 1.75·24-s − 10.4·26-s + 27-s + 4.98·28-s + 2.57·29-s + 5.71·31-s − 5.12·32-s − 5.92·33-s − 7.01·34-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.491·4-s + 0.705·6-s + 1.91·7-s − 0.621·8-s + 0.333·9-s − 1.78·11-s + 0.283·12-s − 1.67·13-s + 2.33·14-s − 1.24·16-s − 0.984·17-s + 0.407·18-s + 0.528·19-s + 1.10·21-s − 2.18·22-s − 0.791·23-s − 0.358·24-s − 2.04·26-s + 0.192·27-s + 0.941·28-s + 0.477·29-s + 1.02·31-s − 0.905·32-s − 1.03·33-s − 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 - 1.72T + 2T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 + 5.92T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 - 5.71T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 5.60T + 43T^{2} \) |
| 47 | \( 1 + 3.92T + 47T^{2} \) |
| 53 | \( 1 - 0.0883T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 6.15T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 - 0.200T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 3.83T + 83T^{2} \) |
| 89 | \( 1 - 2.21T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.53082994878987116109559469556, −6.84583526696736493373551274176, −5.71903989399730042917373196908, −5.09291603575588692928054981276, −4.67073530332942168479151795021, −4.29866201813303948409977399029, −2.92289601815080405038017469422, −2.54833163276147734170503889484, −1.75243284688782541046029943005, 0,
1.75243284688782541046029943005, 2.54833163276147734170503889484, 2.92289601815080405038017469422, 4.29866201813303948409977399029, 4.67073530332942168479151795021, 5.09291603575588692928054981276, 5.71903989399730042917373196908, 6.84583526696736493373551274176, 7.53082994878987116109559469556
