L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2·13-s − 14-s + 16-s + 6·17-s − 19-s + 6·23-s + 2·26-s − 28-s − 6·29-s + 5·31-s + 32-s + 6·34-s − 7·37-s − 38-s + 12·41-s + 11·43-s + 6·46-s + 12·47-s − 6·49-s + 2·52-s − 56-s − 6·58-s − 6·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.229·19-s + 1.25·23-s + 0.392·26-s − 0.188·28-s − 1.11·29-s + 0.898·31-s + 0.176·32-s + 1.02·34-s − 1.15·37-s − 0.162·38-s + 1.87·41-s + 1.67·43-s + 0.884·46-s + 1.75·47-s − 6/7·49-s + 0.277·52-s − 0.133·56-s − 0.787·58-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.700204532\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.700204532\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.597258108835346500353187268680, −8.877507904259066703186689727633, −7.75955926681846106813868928019, −7.15099220962857184034307251192, −6.06893672802395610208258293619, −5.54847638814187101986511829563, −4.43927097888500795100340187556, −3.52615764867254651699567926692, −2.67580284479230715758367028282, −1.17493906369860869798738419881,
1.17493906369860869798738419881, 2.67580284479230715758367028282, 3.52615764867254651699567926692, 4.43927097888500795100340187556, 5.54847638814187101986511829563, 6.06893672802395610208258293619, 7.15099220962857184034307251192, 7.75955926681846106813868928019, 8.877507904259066703186689727633, 9.597258108835346500353187268680