| L(s) = 1 | − 5.12·2-s − 3·3-s + 18.3·4-s + 15.2·5-s + 15.3·6-s + 8.97·7-s − 52.8·8-s + 9·9-s − 78.2·10-s + 2.67·11-s − 54.9·12-s + 17.6·13-s − 46.0·14-s − 45.7·15-s + 124.·16-s + 44.5·17-s − 46.1·18-s − 74.5·19-s + 279.·20-s − 26.9·21-s − 13.7·22-s + 114.·23-s + 158.·24-s + 107.·25-s − 90.7·26-s − 27·27-s + 164.·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.28·4-s + 1.36·5-s + 1.04·6-s + 0.484·7-s − 2.33·8-s + 0.333·9-s − 2.47·10-s + 0.0734·11-s − 1.32·12-s + 0.377·13-s − 0.878·14-s − 0.787·15-s + 1.94·16-s + 0.635·17-s − 0.604·18-s − 0.900·19-s + 3.12·20-s − 0.279·21-s − 0.133·22-s + 1.03·23-s + 1.34·24-s + 0.861·25-s − 0.684·26-s − 0.192·27-s + 1.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.078284397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078284397\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 241 | \( 1 + 241T \) |
| good | 2 | \( 1 + 5.12T + 8T^{2} \) |
| 5 | \( 1 - 15.2T + 125T^{2} \) |
| 7 | \( 1 - 8.97T + 343T^{2} \) |
| 11 | \( 1 - 2.67T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 51.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 114.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 177.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 311.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 167.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 299.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 466.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 30.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 731.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 14.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 364.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 652.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 841.T + 9.12e5T^{2} \) |
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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
−10.12371106474742264554957030259, −9.173425266281441621728535967279, −8.531595521118081413037558071455, −7.56936717189602211637073169537, −6.51024772878169992062678562352, −6.04369543503585217474351120399, −4.81741687479213758858404825089, −2.75883873204361608111362338106, −1.67098194763521228621120999107, −0.859732002844405534054728134781,
0.859732002844405534054728134781, 1.67098194763521228621120999107, 2.75883873204361608111362338106, 4.81741687479213758858404825089, 6.04369543503585217474351120399, 6.51024772878169992062678562352, 7.56936717189602211637073169537, 8.531595521118081413037558071455, 9.173425266281441621728535967279, 10.12371106474742264554957030259
