| L(s) = 1 | + 1.07i·5-s − 7.21·7-s + 16.3i·11-s + 21.6·13-s − 18.9i·17-s − 17.0·19-s + 1.11i·23-s + 23.8·25-s + 29.4i·29-s + 5.63·31-s − 7.75i·35-s − 17.0·37-s − 27.4i·41-s + 52.3·43-s + 64.5i·47-s + ⋯ |
| L(s) = 1 | + 0.214i·5-s − 1.03·7-s + 1.48i·11-s + 1.66·13-s − 1.11i·17-s − 0.897·19-s + 0.0485i·23-s + 0.953·25-s + 1.01i·29-s + 0.181·31-s − 0.221i·35-s − 0.460·37-s − 0.669i·41-s + 1.21·43-s + 1.37i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.161297983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.161297983\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.07iT - 25T^{2} \) |
| 7 | \( 1 + 7.21T + 49T^{2} \) |
| 11 | \( 1 - 16.3iT - 121T^{2} \) |
| 13 | \( 1 - 21.6T + 169T^{2} \) |
| 17 | \( 1 + 18.9iT - 289T^{2} \) |
| 19 | \( 1 + 17.0T + 361T^{2} \) |
| 23 | \( 1 - 1.11iT - 529T^{2} \) |
| 29 | \( 1 - 29.4iT - 841T^{2} \) |
| 31 | \( 1 - 5.63T + 961T^{2} \) |
| 37 | \( 1 + 17.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 27.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 52.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 64.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 56.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 69.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 69.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 98.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 37.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 7.75iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 76.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 4.84T + 9.40e3T^{2} \) |
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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
−9.162834072826529703581288921982, −8.452220607271828788230457496768, −7.34273636722267125417245187007, −6.77707829894069428396760163706, −6.16842611837131477637113235539, −5.10817516788247715641942262972, −4.20559552233562146041671816053, −3.33620616025155586720763136514, −2.42803518349022408245705235083, −1.17392278174882214030952800243,
0.31108377773153312010047639246, 1.39853560623343096381039132576, 2.86355890642751310533398444241, 3.60875395311379171548218955589, 4.31489720025688158917562369039, 5.82047239414471394329819782291, 6.05948140759756275062589294813, 6.77146768828776108610294777773, 8.053985285573752371520685856845, 8.632042782282380065063880707949
