| L(s) = 1 | − 2.40·3-s + 4.41·7-s + 2.76·9-s − 2.29·11-s − 6.92·13-s − 1.51·17-s − 2.89·19-s − 10.5·21-s + 23-s + 0.570·27-s − 7.68·29-s − 3.85·31-s + 5.50·33-s − 8.62·37-s + 16.6·39-s − 6.44·41-s − 3.48·43-s − 6.19·47-s + 12.4·49-s + 3.63·51-s − 2.17·53-s + 6.95·57-s + 11.7·59-s − 5.11·61-s + 12.1·63-s + 9.94·67-s − 2.40·69-s + ⋯ |
| L(s) = 1 | − 1.38·3-s + 1.66·7-s + 0.920·9-s − 0.691·11-s − 1.92·13-s − 0.367·17-s − 0.665·19-s − 2.31·21-s + 0.208·23-s + 0.109·27-s − 1.42·29-s − 0.692·31-s + 0.958·33-s − 1.41·37-s + 2.66·39-s − 1.00·41-s − 0.531·43-s − 0.903·47-s + 1.78·49-s + 0.508·51-s − 0.299·53-s + 0.921·57-s + 1.53·59-s − 0.654·61-s + 1.53·63-s + 1.21·67-s − 0.288·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6157548183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6157548183\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.40T + 3T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 6.92T + 13T^{2} \) |
| 17 | \( 1 + 1.51T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 + 3.48T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 + 2.17T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 - 8.95T + 73T^{2} \) |
| 79 | \( 1 - 1.92T + 79T^{2} \) |
| 83 | \( 1 - 8.04T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.50166561422393028760200103527, −7.14597739273256723861173935694, −6.29120812047894954281182527773, −5.33870694848364936551149051347, −5.00800403331143199090315381428, −4.75363376381643497742391835610, −3.62152737903960048227270136791, −2.25177309458574656775800964652, −1.78984357300340478769038787161, −0.39224819958003626471965269354,
0.39224819958003626471965269354, 1.78984357300340478769038787161, 2.25177309458574656775800964652, 3.62152737903960048227270136791, 4.75363376381643497742391835610, 5.00800403331143199090315381428, 5.33870694848364936551149051347, 6.29120812047894954281182527773, 7.14597739273256723861173935694, 7.50166561422393028760200103527
