| L(s) = 1 | + 3.37·2-s + 3.42·4-s + 5·5-s − 15.4·8-s + 16.8·10-s − 45.0·11-s + 35.4·13-s − 79.6·16-s + 29.4·17-s − 3.18·19-s + 17.1·20-s − 152.·22-s + 23.6·23-s + 25·25-s + 119.·26-s − 9.22·29-s + 80.2·31-s − 145.·32-s + 99.5·34-s − 61.1·37-s − 10.7·38-s − 77.3·40-s + 282.·41-s − 58.8·43-s − 154.·44-s + 79.9·46-s + 371.·47-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 0.427·4-s + 0.447·5-s − 0.683·8-s + 0.534·10-s − 1.23·11-s + 0.757·13-s − 1.24·16-s + 0.420·17-s − 0.0384·19-s + 0.191·20-s − 1.47·22-s + 0.214·23-s + 0.200·25-s + 0.904·26-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 − 0.305·40-s + 1.07·41-s − 0.208·43-s − 0.528·44-s + 0.256·46-s + 1.15·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.803400990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.803400990\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| 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
−8.685218082013677402537399599875, −7.88191357809470810442507506766, −6.91037716455381370702497065792, −6.00675172207390754665643968875, −5.50010578978888437905216517563, −4.77303868269411399712601778807, −3.86210442597502143916945796655, −2.99899136855348031529262329656, −2.20230201449481833265029316150, −0.71934658620606453560218725905,
0.71934658620606453560218725905, 2.20230201449481833265029316150, 2.99899136855348031529262329656, 3.86210442597502143916945796655, 4.77303868269411399712601778807, 5.50010578978888437905216517563, 6.00675172207390754665643968875, 6.91037716455381370702497065792, 7.88191357809470810442507506766, 8.685218082013677402537399599875
