| L(s) = 1 | + 2.84·2-s − 4.64·3-s + 0.120·4-s + 12.8·5-s − 13.2·6-s + 26.0·7-s − 22.4·8-s − 5.41·9-s + 36.6·10-s + 62.8·11-s − 0.561·12-s + 22.3·13-s + 74.2·14-s − 59.8·15-s − 64.9·16-s + 57.9·17-s − 15.4·18-s − 71.3·19-s + 1.55·20-s − 121.·21-s + 179.·22-s − 49.5·23-s + 104.·24-s + 40.7·25-s + 63.8·26-s + 150.·27-s + 3.15·28-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.894·3-s + 0.0151·4-s + 1.15·5-s − 0.900·6-s + 1.40·7-s − 0.992·8-s − 0.200·9-s + 1.16·10-s + 1.72·11-s − 0.0135·12-s + 0.477·13-s + 1.41·14-s − 1.02·15-s − 1.01·16-s + 0.827·17-s − 0.202·18-s − 0.861·19-s + 0.0174·20-s − 1.25·21-s + 1.73·22-s − 0.449·23-s + 0.887·24-s + 0.325·25-s + 0.481·26-s + 1.07·27-s + 0.0212·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.524874979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.524874979\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 2.84T + 8T^{2} \) |
| 3 | \( 1 + 4.64T + 27T^{2} \) |
| 5 | \( 1 - 12.8T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 11 | \( 1 - 62.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.5T + 1.21e4T^{2} \) |
| 31 | \( 1 + 62.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 553.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 136.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 579.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 919.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 781.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 133.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 868.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 83.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 357.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 187.T + 9.12e5T^{2} \) |
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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
−9.843071603924402914158566924414, −9.017294720359769869226300245500, −8.227114696106398436273415767447, −6.66826278171911509584435690890, −5.99270908851123597909590707586, −5.46582327877107365270020372093, −4.60117457682736770760175663377, −3.72030523243440736886775839187, −2.11004858194931606521321828647, −1.00854614133414861905743179698,
1.00854614133414861905743179698, 2.11004858194931606521321828647, 3.72030523243440736886775839187, 4.60117457682736770760175663377, 5.46582327877107365270020372093, 5.99270908851123597909590707586, 6.66826278171911509584435690890, 8.227114696106398436273415767447, 9.017294720359769869226300245500, 9.843071603924402914158566924414
