| L(s) = 1 | + 1.97i·5-s − 1.97i·7-s + 11-s − 6.65·13-s + 5.31i·17-s − 4.53i·19-s + 6.54·23-s + 1.10·25-s + 3.46i·29-s − 8.77i·31-s + 3.89·35-s + 2.72·37-s − 8.77i·41-s + 9.07i·43-s − 13.3·47-s + ⋯ |
| L(s) = 1 | + 0.882i·5-s − 0.745i·7-s + 0.301·11-s − 1.84·13-s + 1.28i·17-s − 1.04i·19-s + 1.36·23-s + 0.221·25-s + 0.643i·29-s − 1.57i·31-s + 0.657·35-s + 0.447·37-s − 1.37i·41-s + 1.38i·43-s − 1.95·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.597877609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597877609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 1.97iT - 5T^{2} \) |
| 7 | \( 1 + 1.97iT - 7T^{2} \) |
| 13 | \( 1 + 6.65T + 13T^{2} \) |
| 17 | \( 1 - 5.31iT - 17T^{2} \) |
| 19 | \( 1 + 4.53iT - 19T^{2} \) |
| 23 | \( 1 - 6.54T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 8.77iT - 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 + 8.77iT - 41T^{2} \) |
| 43 | \( 1 - 9.07iT - 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 4.95iT - 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 5.02iT - 67T^{2} \) |
| 71 | \( 1 - 6.54T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 - 8.65iT - 79T^{2} \) |
| 83 | \( 1 - 8.10T + 83T^{2} \) |
| 89 | \( 1 + 6.63iT - 89T^{2} \) |
| 97 | \( 1 - 0.109T + 97T^{2} \) |
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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
−7.913331085023874919892775030501, −7.32206148303403867568336765745, −6.85557178520365029458470350835, −6.23970721010454944384217033477, −5.15245464186987752360395793771, −4.56182399893169740898053674018, −3.69746633146232516145152272616, −2.85823345592559654608029805881, −2.15983456637838546572501849317, −0.825812954928076789593491977774,
0.51891173725560737107263238274, 1.68471671502702149391761144999, 2.63362607431833507333600951037, 3.35237253735528163728399591962, 4.71573275096161889061524223254, 4.92444773761241637533134833562, 5.56162317348461187371983278760, 6.64705098528812016046189653822, 7.17844547590474702025947167092, 8.024387567639097336200870997441
