L(s) = 1 | + 2-s + 4-s − 3·5-s − 7-s + 8-s − 3·10-s − 3·13-s − 14-s + 16-s + 17-s + 6·19-s − 3·20-s + 2·23-s + 4·25-s − 3·26-s − 28-s + 6·29-s + 32-s + 34-s + 3·35-s + 3·37-s + 6·38-s − 3·40-s − 11·43-s + 2·46-s + 49-s + 4·50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 0.670·20-s + 0.417·23-s + 4/5·25-s − 0.588·26-s − 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.171·34-s + 0.507·35-s + 0.493·37-s + 0.973·38-s − 0.474·40-s − 1.67·43-s + 0.294·46-s + 1/7·49-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.08695505465439, −12.41407625468298, −12.08315753908658, −11.81258999337673, −11.44519618140248, −10.89900764755875, −10.28369173950676, −9.896547456238555, −9.462864599422784, −8.722369012770987, −8.259935040964131, −7.836529852348362, −7.279381240025369, −6.985983583076933, −6.563827117096623, −5.763184006282556, −5.303610958457061, −4.850618125647826, −4.339985732753259, −3.790766645559409, −3.347784499347987, −2.848102040729496, −2.387741311190726, −1.378984634285586, −0.7841780862970768, 0,
0.7841780862970768, 1.378984634285586, 2.387741311190726, 2.848102040729496, 3.347784499347987, 3.790766645559409, 4.339985732753259, 4.850618125647826, 5.303610958457061, 5.763184006282556, 6.563827117096623, 6.985983583076933, 7.279381240025369, 7.836529852348362, 8.259935040964131, 8.722369012770987, 9.462864599422784, 9.896547456238555, 10.28369173950676, 10.89900764755875, 11.44519618140248, 11.81258999337673, 12.08315753908658, 12.41407625468298, 13.08695505465439