| L(s) = 1 | − i·7-s − 4i·13-s + 6i·17-s − 2·19-s − 6i·23-s + 2·31-s − 2i·37-s − 6·41-s − 4i·43-s − 49-s + 6i·53-s + 12·59-s − 10·61-s + 4i·67-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.377i·7-s − 1.10i·13-s + 1.45i·17-s − 0.458·19-s − 1.25i·23-s + 0.359·31-s − 0.328i·37-s − 0.937·41-s − 0.609i·43-s − 0.142·49-s + 0.824i·53-s + 1.56·59-s − 1.28·61-s + 0.488i·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6991260109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6991260109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−7.79584653969953619602081406860, −7.05668926364497742611776494818, −6.24406772921903899852710795949, −5.74437763488870429412853053691, −4.77910875495351423081792841561, −4.09142190210313933200620933421, −3.30408898715098432064263981032, −2.40333385791447483184842600513, −1.35870845172254980656310165912, −0.17588756095700603075518696620,
1.30606241297283033813531609135, 2.25583481823764720880552468080, 3.09010947827642034964368583757, 3.99144249593337090452951289436, 4.82220770363466096017042042723, 5.39651702837172225646394531581, 6.32657597080790702260491555873, 6.91426317433602509382078626176, 7.56915523152940761542252221423, 8.382100079990545844759016527139
