| L(s) = 1 | − 30.5·3-s + 4.01e3·7-s − 1.87e4·9-s + 4.21e4·11-s + 1.23e5·13-s + 3.19e5·17-s − 1.08e6·19-s − 1.22e5·21-s − 1.50e6·23-s + 1.17e6·27-s − 2.62e6·29-s − 3.27e6·31-s − 1.28e6·33-s − 2.51e6·37-s − 3.77e6·39-s + 2.95e7·41-s + 1.42e7·43-s − 1.35e6·47-s − 2.42e7·49-s − 9.76e6·51-s + 9.73e7·53-s + 3.31e7·57-s + 7.48e6·59-s − 9.11e7·61-s − 7.52e7·63-s − 2.94e8·67-s + 4.59e7·69-s + ⋯ |
| L(s) = 1 | − 0.217·3-s + 0.631·7-s − 0.952·9-s + 0.867·11-s + 1.20·13-s + 0.929·17-s − 1.91·19-s − 0.137·21-s − 1.12·23-s + 0.424·27-s − 0.688·29-s − 0.635·31-s − 0.188·33-s − 0.220·37-s − 0.261·39-s + 1.63·41-s + 0.635·43-s − 0.0404·47-s − 0.601·49-s − 0.202·51-s + 1.69·53-s + 0.416·57-s + 0.0804·59-s − 0.843·61-s − 0.601·63-s − 1.78·67-s + 0.244·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 30.5T + 1.96e4T^{2} \) |
| 7 | \( 1 - 4.01e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.21e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.23e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.19e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.08e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.50e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.51e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.95e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.35e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.73e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.48e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.11e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.94e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.82e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.48e6T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.25e8T + 7.60e17T^{2} \) |
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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
−9.068780995190921582844332468672, −8.507193459332937899851008712978, −7.56782408312553048388910504912, −6.17831575250707882051588714239, −5.80433075120218912925796937517, −4.38758986166924061250061789829, −3.58318522449777500164746017189, −2.19221982310895030789353597593, −1.20238427993438059216118919276, 0,
1.20238427993438059216118919276, 2.19221982310895030789353597593, 3.58318522449777500164746017189, 4.38758986166924061250061789829, 5.80433075120218912925796937517, 6.17831575250707882051588714239, 7.56782408312553048388910504912, 8.507193459332937899851008712978, 9.068780995190921582844332468672
