| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 4·11-s − 13-s + 4·14-s + 16-s + 2·17-s − 8·19-s − 4·22-s + 26-s − 4·28-s − 6·29-s − 4·31-s − 32-s − 2·34-s + 2·37-s + 8·38-s + 10·41-s − 4·43-s + 4·44-s + 8·47-s + 9·49-s − 52-s − 10·53-s + 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 1.20·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 1.83·19-s − 0.852·22-s + 0.196·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s + 0.328·37-s + 1.29·38-s + 1.56·41-s − 0.609·43-s + 0.603·44-s + 1.16·47-s + 9/7·49-s − 0.138·52-s − 1.37·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8217585577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8217585577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.151429755757169983694191330601, −7.37332627467142708071248165957, −6.58301651034150370973585285363, −6.31913435235909204329849344713, −5.48757254823403234741415051096, −4.16221641487234182394243329902, −3.66102336194150189960878165059, −2.69591224070858700372522956664, −1.79208018239078151901896144058, −0.51910506533247467964771569864,
0.51910506533247467964771569864, 1.79208018239078151901896144058, 2.69591224070858700372522956664, 3.66102336194150189960878165059, 4.16221641487234182394243329902, 5.48757254823403234741415051096, 6.31913435235909204329849344713, 6.58301651034150370973585285363, 7.37332627467142708071248165957, 8.151429755757169983694191330601
