| L(s) = 1 | + (−1 − 2i)5-s + i·7-s − 6·11-s + 2i·13-s − 4i·17-s − 6·19-s + (−3 + 4i)25-s − 2·29-s + 10·31-s + (2 − i)35-s − 4i·37-s − 2·41-s − 4i·43-s − 49-s + 6i·53-s + ⋯ |
| L(s) = 1 | + (−0.447 − 0.894i)5-s + 0.377i·7-s − 1.80·11-s + 0.554i·13-s − 0.970i·17-s − 1.37·19-s + (−0.600 + 0.800i)25-s − 0.371·29-s + 1.79·31-s + (0.338 − 0.169i)35-s − 0.657i·37-s − 0.312·41-s − 0.609i·43-s − 0.142·49-s + 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9481000253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9481000253\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.361835118147274635800141074582, −7.69079784765016016213972563289, −6.97403708547201024351216230581, −5.99481893428776636770374903919, −5.23252694421221913589755940860, −4.71565485860379540356833093134, −3.94985970118606500879049095822, −2.75377545149421420779722922875, −2.11936001204954947106587096675, −0.66703873303515375556556336169,
0.38284125303428490294899336549, 2.05209531792832831436100079288, 2.81268314501469923363982832570, 3.57058368523390818353692304276, 4.46823829278461423382501114175, 5.21264409329943750214104591463, 6.23338218497328951643090127303, 6.63075326421363053411211455733, 7.70625668077413226833832602500, 8.019167932013457252481775158744
