L(s) = 1 | − 3-s − 4·5-s − 7-s − 2·9-s + 3·11-s − 3·13-s + 4·15-s − 4·17-s + 21-s + 3·23-s + 11·25-s + 5·27-s − 10·29-s − 10·31-s − 3·33-s + 4·35-s + 2·37-s + 3·39-s + 2·41-s − 5·43-s + 8·45-s − 11·47-s − 6·49-s + 4·51-s − 14·53-s − 12·55-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.832·13-s + 1.03·15-s − 0.970·17-s + 0.218·21-s + 0.625·23-s + 11/5·25-s + 0.962·27-s − 1.85·29-s − 1.79·31-s − 0.522·33-s + 0.676·35-s + 0.328·37-s + 0.480·39-s + 0.312·41-s − 0.762·43-s + 1.19·45-s − 1.60·47-s − 6/7·49-s + 0.560·51-s − 1.92·53-s − 1.61·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.05853910617118554580534790727, −6.66684212083147676357063911266, −5.73187463604685061945618102138, −4.91515002926748108152492475912, −4.30608580796186110467344058590, −3.53432281850550197267373253901, −2.96765131608937582117178318141, −1.61961333645619530687010969932, 0, 0,
1.61961333645619530687010969932, 2.96765131608937582117178318141, 3.53432281850550197267373253901, 4.30608580796186110467344058590, 4.91515002926748108152492475912, 5.73187463604685061945618102138, 6.66684212083147676357063911266, 7.05853910617118554580534790727