L(s) = 1 | − 0.805·2-s − 2.37·3-s − 1.35·4-s + 0.559·5-s + 1.91·6-s + 1.71·7-s + 2.69·8-s + 2.63·9-s − 0.450·10-s + 3.20·12-s + 13-s − 1.37·14-s − 1.32·15-s + 0.529·16-s + 3.42·17-s − 2.12·18-s + 1.18·19-s − 0.755·20-s − 4.06·21-s − 5.43·23-s − 6.40·24-s − 4.68·25-s − 0.805·26-s + 0.862·27-s − 2.31·28-s + 1.37·29-s + 1.06·30-s + ⋯ |
L(s) = 1 | − 0.569·2-s − 1.37·3-s − 0.675·4-s + 0.250·5-s + 0.780·6-s + 0.646·7-s + 0.954·8-s + 0.878·9-s − 0.142·10-s + 0.926·12-s + 0.277·13-s − 0.368·14-s − 0.342·15-s + 0.132·16-s + 0.831·17-s − 0.500·18-s + 0.271·19-s − 0.168·20-s − 0.886·21-s − 1.13·23-s − 1.30·24-s − 0.937·25-s − 0.157·26-s + 0.166·27-s − 0.437·28-s + 0.255·29-s + 0.195·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6342324709\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6342324709\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.805T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 0.559T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 19 | \( 1 - 1.18T + 19T^{2} \) |
| 23 | \( 1 + 5.43T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 - 4.36T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 1.24T + 53T^{2} \) |
| 59 | \( 1 + 1.13T + 59T^{2} \) |
| 61 | \( 1 - 6.98T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 8.97T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 7.92T + 79T^{2} \) |
| 83 | \( 1 + 1.55T + 83T^{2} \) |
| 89 | \( 1 - 0.645T + 89T^{2} \) |
| 97 | \( 1 - 0.0922T + 97T^{2} \) |
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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
−9.707076569199888245973845470473, −8.510263903668830885577397831859, −8.015770698517045305842008125528, −7.01835282352260986866068250356, −6.00659049496244848372260436132, −5.34993777826559634819216007324, −4.67961280527478523184178126905, −3.66828153616400165786136436011, −1.78880560403159622892627927272, −0.68276748547992968533617051069,
0.68276748547992968533617051069, 1.78880560403159622892627927272, 3.66828153616400165786136436011, 4.67961280527478523184178126905, 5.34993777826559634819216007324, 6.00659049496244848372260436132, 7.01835282352260986866068250356, 8.015770698517045305842008125528, 8.510263903668830885577397831859, 9.707076569199888245973845470473