| L(s) = 1 | + 1.07e3i·5-s + (−436. + 2.36e3i)7-s − 1.05e4·11-s − 2.46e4i·13-s + 6.36e4i·17-s + 1.36e5i·19-s − 3.40e5·23-s − 7.57e5·25-s − 9.67e5·29-s − 9.44e4i·31-s + (−2.53e6 − 4.68e5i)35-s + 7.33e5·37-s + 2.00e6i·41-s + 5.98e6·43-s + 1.51e6i·47-s + ⋯ |
| L(s) = 1 | + 1.71i·5-s + (−0.182 + 0.983i)7-s − 0.722·11-s − 0.863i·13-s + 0.762i·17-s + 1.04i·19-s − 1.21·23-s − 1.93·25-s − 1.36·29-s − 0.102i·31-s + (−1.68 − 0.312i)35-s + 0.391·37-s + 0.710i·41-s + 1.74·43-s + 0.310i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.4215326319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4215326319\) |
| \(L(5)\) |
|
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 | \( 1 + (436. - 2.36e3i)T \) |
| good | 5 | \( 1 - 1.07e3iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 1.05e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 2.46e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 6.36e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.36e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 3.40e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 9.67e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 9.44e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 7.33e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.00e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.98e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.51e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 3.99e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 9.18e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.02e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 4.55e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.39e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 3.93e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 7.44e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 1.51e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.77e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 8.92e7iT - 7.83e15T^{2} \) |
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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
−11.11094530832127936909270995507, −10.41454554717521510167284166882, −9.639923914814447257622854979488, −8.165840859246691899612524073468, −7.49075958299576880839680357144, −6.12805045747135403897830240431, −5.70214287152999003348041742133, −3.81419561873638821797183211335, −2.84913107140573108905384344038, −2.01412476921096773715522747760,
0.10505375140536629555119970275, 0.887704086759714670512053231976, 2.15277758201546996670079918881, 3.93457728112961058103138872804, 4.68933396113775986671570686420, 5.65644753873051424696068912820, 7.10950736822046135390032692672, 7.986918939151639778474288438240, 9.078169872465811708606933192112, 9.677825467821188114698425244404
