| L(s) = 1 | + 5.39·2-s + 21.1·4-s + 13.0·5-s + 6.42·7-s + 70.7·8-s + 70.3·10-s + 26.2·11-s + 34.6·14-s + 212.·16-s + 123.·17-s − 109.·19-s + 275.·20-s + 141.·22-s + 63.4·23-s + 45.0·25-s + 135.·28-s − 225.·29-s − 200.·31-s + 582.·32-s + 668.·34-s + 83.7·35-s − 252.·37-s − 591.·38-s + 922.·40-s + 227.·41-s − 384.·43-s + 553.·44-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.63·4-s + 1.16·5-s + 0.346·7-s + 3.12·8-s + 2.22·10-s + 0.718·11-s + 0.661·14-s + 3.32·16-s + 1.76·17-s − 1.32·19-s + 3.07·20-s + 1.37·22-s + 0.574·23-s + 0.360·25-s + 0.915·28-s − 1.44·29-s − 1.16·31-s + 3.21·32-s + 3.37·34-s + 0.404·35-s − 1.12·37-s − 2.52·38-s + 3.64·40-s + 0.866·41-s − 1.36·43-s + 1.89·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.25877299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.25877299\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 5.39T + 8T^{2} \) |
| 5 | \( 1 - 13.0T + 125T^{2} \) |
| 7 | \( 1 - 6.42T + 343T^{2} \) |
| 11 | \( 1 - 26.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 63.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 200.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 384.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 34.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 61.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 80.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 26.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 931.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 427.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 108.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 384.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 85.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 495.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 190.T + 9.12e5T^{2} \) |
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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
−9.259794048686754841581035027074, −8.006553822553417236210869113732, −7.04590747564720413202556417015, −6.35659402088071693104973217586, −5.55624164633518860581543060571, −5.16837712529874786886472182064, −3.99779433825254292028315153284, −3.30591493566974590298148176030, −2.09718414329780668360003149511, −1.50333928784680741044706402142,
1.50333928784680741044706402142, 2.09718414329780668360003149511, 3.30591493566974590298148176030, 3.99779433825254292028315153284, 5.16837712529874786886472182064, 5.55624164633518860581543060571, 6.35659402088071693104973217586, 7.04590747564720413202556417015, 8.006553822553417236210869113732, 9.259794048686754841581035027074
