L(s) = 1 | − 2·5-s − 7-s + 2·13-s − 4·17-s − 4·19-s + 6·23-s − 25-s − 2·29-s − 2·31-s + 2·35-s + 2·37-s − 6·43-s + 12·47-s + 49-s + 12·59-s − 2·61-s − 4·65-s + 4·67-s + 6·71-s + 10·73-s − 16·79-s − 6·83-s + 8·85-s + 6·89-s − 2·91-s + 8·95-s − 6·97-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.554·13-s − 0.970·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s − 0.371·29-s − 0.359·31-s + 0.338·35-s + 0.328·37-s − 0.914·43-s + 1.75·47-s + 1/7·49-s + 1.56·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.712·71-s + 1.17·73-s − 1.80·79-s − 0.658·83-s + 0.867·85-s + 0.635·89-s − 0.209·91-s + 0.820·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.019371594\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.019371594\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.37297072927481, −13.64801444158077, −13.25017932353968, −12.80373335771785, −12.33925689923017, −11.67675314227985, −11.19302465984713, −10.90976202809078, −10.31118695627771, −9.635716986889130, −9.035931308123467, −8.623677526038683, −8.167608079719018, −7.486844974138103, −6.917205399290138, −6.589527427642484, −5.840824394347271, −5.258179020804662, −4.527618059967795, −3.952568354272269, −3.629129515578905, −2.750087649170404, −2.193460246594804, −1.235401298611352, −0.3638282632530825,
0.3638282632530825, 1.235401298611352, 2.193460246594804, 2.750087649170404, 3.629129515578905, 3.952568354272269, 4.527618059967795, 5.258179020804662, 5.840824394347271, 6.589527427642484, 6.917205399290138, 7.486844974138103, 8.167608079719018, 8.623677526038683, 9.035931308123467, 9.635716986889130, 10.31118695627771, 10.90976202809078, 11.19302465984713, 11.67675314227985, 12.33925689923017, 12.80373335771785, 13.25017932353968, 13.64801444158077, 14.37297072927481