| L(s) = 1 | − 3·3-s + 7·7-s + 9·9-s + 47.8·11-s + 89.2·13-s − 3.20·17-s + 114.·19-s − 21·21-s + 196.·23-s − 27·27-s + 176.·29-s + 3.59·31-s − 143.·33-s + 308.·37-s − 267.·39-s − 179.·41-s − 374.·43-s + 405.·47-s + 49·49-s + 9.60·51-s − 243.·53-s − 344.·57-s − 91.1·59-s + 172.·61-s + 63·63-s + 373.·67-s − 588.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 0.333·9-s + 1.31·11-s + 1.90·13-s − 0.0456·17-s + 1.38·19-s − 0.218·21-s + 1.77·23-s − 0.192·27-s + 1.12·29-s + 0.0208·31-s − 0.757·33-s + 1.37·37-s − 1.09·39-s − 0.683·41-s − 1.32·43-s + 1.25·47-s + 0.142·49-s + 0.0263·51-s − 0.632·53-s − 0.801·57-s − 0.201·59-s + 0.361·61-s + 0.125·63-s + 0.680·67-s − 1.02·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.936747449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.936747449\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 11 | \( 1 - 47.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 89.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.20T + 4.91e3T^{2} \) |
| 19 | \( 1 - 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 196.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 3.59T + 2.97e4T^{2} \) |
| 37 | \( 1 - 308.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 405.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 243.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 172.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 373.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 609.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 514.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 866.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 309.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 277.T + 9.12e5T^{2} \) |
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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
−8.839020637679438962438054246907, −8.028316523365771278888743096194, −6.97584779158917003164441571871, −6.43505295599236003113349769683, −5.63016141333174865729898076790, −4.75833103936539815706254760731, −3.86616723306911510718976498766, −3.03231072742430237417772370014, −1.32552411364965075653767605229, −1.00949073374241936151229301270,
1.00949073374241936151229301270, 1.32552411364965075653767605229, 3.03231072742430237417772370014, 3.86616723306911510718976498766, 4.75833103936539815706254760731, 5.63016141333174865729898076790, 6.43505295599236003113349769683, 6.97584779158917003164441571871, 8.028316523365771278888743096194, 8.839020637679438962438054246907
