L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 4·13-s + 14-s + 16-s − 6·17-s + 2·19-s − 5·25-s − 4·26-s + 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s + 2·37-s + 2·38-s − 6·41-s + 8·43-s + 12·47-s + 49-s − 5·50-s − 4·52-s − 6·53-s + 56-s + 6·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.707·50-s − 0.554·52-s − 0.824·53-s + 0.133·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.530545448\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.530545448\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.45852272915611541668421078683, −12.37171571172372853840948077604, −11.52629384186915375545261187858, −10.47027853370113893616936981615, −9.207333736191082013174831123730, −7.80365099617056766338064266882, −6.71310127772875365703449938682, −5.31921059379060528260176389250, −4.19102102405251164216473944262, −2.39101862347891857041372548490,
2.39101862347891857041372548490, 4.19102102405251164216473944262, 5.31921059379060528260176389250, 6.71310127772875365703449938682, 7.80365099617056766338064266882, 9.207333736191082013174831123730, 10.47027853370113893616936981615, 11.52629384186915375545261187858, 12.37171571172372853840948077604, 13.45852272915611541668421078683