| L(s) = 1 | + 3.37·2-s + 3.42·4-s − 7·7-s − 15.4·8-s + 45.0·11-s + 35.4·13-s − 23.6·14-s − 79.6·16-s − 29.4·17-s + 3.18·19-s + 152.·22-s + 23.6·23-s + 119.·26-s − 23.9·28-s + 9.22·29-s − 80.2·31-s − 145.·32-s − 99.5·34-s + 61.1·37-s + 10.7·38-s + 282.·41-s + 58.8·43-s + 154.·44-s + 79.9·46-s − 371.·47-s + 49·49-s + 121.·52-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 0.427·4-s − 0.377·7-s − 0.683·8-s + 1.23·11-s + 0.757·13-s − 0.451·14-s − 1.24·16-s − 0.420·17-s + 0.0384·19-s + 1.47·22-s + 0.214·23-s + 0.904·26-s − 0.161·28-s + 0.0590·29-s − 0.464·31-s − 0.803·32-s − 0.501·34-s + 0.271·37-s + 0.0460·38-s + 1.07·41-s + 0.208·43-s + 0.528·44-s + 0.256·46-s − 1.15·47-s + 0.142·49-s + 0.324·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.785641028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.785641028\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 3.37T + 8T^{2} \) |
| 11 | \( 1 - 45.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 3.18T + 6.85e3T^{2} \) |
| 23 | \( 1 - 23.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 9.22T + 2.43e4T^{2} \) |
| 31 | \( 1 + 80.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 61.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 58.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 371.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 256.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 571.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 835.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 933.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 378.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 494.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 722.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 89.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 101.T + 9.12e5T^{2} \) |
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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
−9.092911380117645912768314239796, −8.385110344560360013366335733840, −7.08967933966741893851128102343, −6.40273606524592762780475405302, −5.79736154047941678764611552868, −4.80138525051682736105827981752, −3.93524044746933462939650295533, −3.41145282001411189093731570427, −2.20886859798885702902309204256, −0.792151043197346274876041187709,
0.792151043197346274876041187709, 2.20886859798885702902309204256, 3.41145282001411189093731570427, 3.93524044746933462939650295533, 4.80138525051682736105827981752, 5.79736154047941678764611552868, 6.40273606524592762780475405302, 7.08967933966741893851128102343, 8.385110344560360013366335733840, 9.092911380117645912768314239796
