| L(s) = 1 | − 3·3-s − 11.4·5-s + 11.2·7-s + 9·9-s − 25.8·11-s + 13·13-s + 34.2·15-s − 20.3·17-s − 154.·19-s − 33.7·21-s + 180.·23-s + 5.69·25-s − 27·27-s − 20.4·29-s − 266.·31-s + 77.6·33-s − 128.·35-s + 115.·37-s − 39·39-s + 391.·41-s − 151.·43-s − 102.·45-s + 467.·47-s − 216.·49-s + 60.9·51-s + 79.9·53-s + 295.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.02·5-s + 0.607·7-s + 0.333·9-s − 0.709·11-s + 0.277·13-s + 0.590·15-s − 0.290·17-s − 1.86·19-s − 0.350·21-s + 1.63·23-s + 0.0455·25-s − 0.192·27-s − 0.130·29-s − 1.54·31-s + 0.409·33-s − 0.621·35-s + 0.515·37-s − 0.160·39-s + 1.49·41-s − 0.536·43-s − 0.340·45-s + 1.45·47-s − 0.630·49-s + 0.167·51-s + 0.207·53-s + 0.725·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.014902422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014902422\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 11.4T + 125T^{2} \) |
| 7 | \( 1 - 11.2T + 343T^{2} \) |
| 11 | \( 1 + 25.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 467.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 79.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 873.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 187.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 609.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 248.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 852.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 331.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 435.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 259.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−10.70702319878318752524067981094, −9.225905207323683479205533910662, −8.351202003037429095931300792865, −7.57668720978151822466820013357, −6.71486153609487080740571710866, −5.54621668960216646944113105638, −4.61169678387003034692640910242, −3.76977568213325285150778648295, −2.22477015589353256171463346938, −0.59978760653450503243651524009,
0.59978760653450503243651524009, 2.22477015589353256171463346938, 3.76977568213325285150778648295, 4.61169678387003034692640910242, 5.54621668960216646944113105638, 6.71486153609487080740571710866, 7.57668720978151822466820013357, 8.351202003037429095931300792865, 9.225905207323683479205533910662, 10.70702319878318752524067981094
