| L(s) = 1 | + 5.38·5-s + 3.38·7-s + 68.9·11-s + 13·13-s + 9.52·17-s − 133.·19-s + 196.·23-s − 96.0·25-s + 4.56·29-s + 254.·31-s + 18.1·35-s + 246.·37-s − 166.·41-s + 422.·43-s + 88.3·47-s − 331.·49-s − 226.·53-s + 370.·55-s + 477.·59-s − 13.2·61-s + 69.9·65-s − 6.99·67-s − 195.·71-s − 652.·73-s + 232.·77-s − 536.·79-s − 1.24e3·83-s + ⋯ |
| L(s) = 1 | + 0.481·5-s + 0.182·7-s + 1.88·11-s + 0.277·13-s + 0.135·17-s − 1.60·19-s + 1.78·23-s − 0.768·25-s + 0.0292·29-s + 1.47·31-s + 0.0878·35-s + 1.09·37-s − 0.633·41-s + 1.49·43-s + 0.274·47-s − 0.966·49-s − 0.587·53-s + 0.908·55-s + 1.05·59-s − 0.0277·61-s + 0.133·65-s − 0.0127·67-s − 0.326·71-s − 1.04·73-s + 0.344·77-s − 0.764·79-s − 1.65·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.074476448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.074476448\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 5.38T + 125T^{2} \) |
| 7 | \( 1 - 3.38T + 343T^{2} \) |
| 11 | \( 1 - 68.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 9.52T + 4.91e3T^{2} \) |
| 19 | \( 1 + 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 196.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 4.56T + 2.43e4T^{2} \) |
| 31 | \( 1 - 254.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 246.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 166.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 88.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 226.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 477.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 13.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 6.99T + 3.00e5T^{2} \) |
| 71 | \( 1 + 195.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 652.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 536.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 370.T + 9.12e5T^{2} \) |
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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
−8.912648666238608395715275536248, −8.281073546339059322219168207294, −7.12739095616712879146210117513, −6.42627984549524220529400092021, −5.89232823700541636274441888206, −4.62262067871413703320465491538, −4.03833329328810967555860041410, −2.88293077683429334838876597402, −1.74138147313860836409829207790, −0.874831035983803976234536379550,
0.874831035983803976234536379550, 1.74138147313860836409829207790, 2.88293077683429334838876597402, 4.03833329328810967555860041410, 4.62262067871413703320465491538, 5.89232823700541636274441888206, 6.42627984549524220529400092021, 7.12739095616712879146210117513, 8.281073546339059322219168207294, 8.912648666238608395715275536248
