L(s) = 1 | − 0.841·5-s + 7-s + 4.29·11-s + 5.02·13-s − 6.86·17-s + 19-s + 3.02·23-s − 4.29·25-s + 2.18·29-s + 7.31·31-s − 0.841·35-s + 3.27·37-s + 0.317·41-s + 12.7·47-s + 49-s − 6.18·53-s − 3.61·55-s − 14.1·59-s + 6·61-s − 4.22·65-s + 1.86·67-s + 7.11·71-s − 2.74·73-s + 4.29·77-s + 1.72·79-s + 6.72·83-s + 5.77·85-s + ⋯ |
L(s) = 1 | − 0.376·5-s + 0.377·7-s + 1.29·11-s + 1.39·13-s − 1.66·17-s + 0.229·19-s + 0.630·23-s − 0.858·25-s + 0.404·29-s + 1.31·31-s − 0.142·35-s + 0.537·37-s + 0.0495·41-s + 1.85·47-s + 0.142·49-s − 0.849·53-s − 0.487·55-s − 1.84·59-s + 0.768·61-s − 0.524·65-s + 0.227·67-s + 0.843·71-s − 0.321·73-s + 0.489·77-s + 0.194·79-s + 0.737·83-s + 0.626·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.398038730\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.398038730\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 0.841T + 5T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 6.86T + 17T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 2.18T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 - 3.27T + 37T^{2} \) |
| 41 | \( 1 - 0.317T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 6.18T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 1.86T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + 2.74T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 - 6.72T + 83T^{2} \) |
| 89 | \( 1 + 3.29T + 89T^{2} \) |
| 97 | \( 1 + 1.43T + 97T^{2} \) |
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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
−7.78758249649793040473142073992, −6.84334460090149493467086134094, −6.41788214412229552699449733916, −5.80747903310820188104155475867, −4.70078021297811194839134753248, −4.20281252449408683236140833305, −3.60599521298646817845975881874, −2.60536847620827166720106523175, −1.60610180456851709967054703843, −0.789532545732488832147249260937,
0.789532545732488832147249260937, 1.60610180456851709967054703843, 2.60536847620827166720106523175, 3.60599521298646817845975881874, 4.20281252449408683236140833305, 4.70078021297811194839134753248, 5.80747903310820188104155475867, 6.41788214412229552699449733916, 6.84334460090149493467086134094, 7.78758249649793040473142073992