| L(s) = 1 | + 3.55·5-s + 7.34·7-s + 9.93·11-s − 7.98i·13-s − 29.0·17-s + (−7.34 + 17.5i)19-s + 10.5·23-s − 12.3·25-s + 35.4i·29-s + 57.2i·31-s + 26.1·35-s + 38.1i·37-s + 9.95i·41-s + 26.0·43-s + 19.9·47-s + ⋯ |
| L(s) = 1 | + 0.711·5-s + 1.04·7-s + 0.903·11-s − 0.613i·13-s − 1.71·17-s + (−0.386 + 0.922i)19-s + 0.458·23-s − 0.493·25-s + 1.22i·29-s + 1.84i·31-s + 0.746·35-s + 1.03i·37-s + 0.242i·41-s + 0.605·43-s + 0.425·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.480327696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480327696\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (7.34 - 17.5i)T \) |
| good | 5 | \( 1 - 3.55T + 25T^{2} \) |
| 7 | \( 1 - 7.34T + 49T^{2} \) |
| 11 | \( 1 - 9.93T + 121T^{2} \) |
| 13 | \( 1 + 7.98iT - 169T^{2} \) |
| 17 | \( 1 + 29.0T + 289T^{2} \) |
| 23 | \( 1 - 10.5T + 529T^{2} \) |
| 29 | \( 1 - 35.4iT - 841T^{2} \) |
| 31 | \( 1 - 57.2iT - 961T^{2} \) |
| 37 | \( 1 - 38.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 9.95iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 26.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 19.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 60.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 79.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 24.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.86iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 62.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 90.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 46.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 65.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 1.74iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.768455953106922883843143553746, −8.215795398792322335592847549631, −7.17722379053400910514522385595, −6.51261570271396890777392520769, −5.70112426392991871657214581795, −4.87908725161624846913226338200, −4.17710889005500713961254711800, −3.03401869190328113616842056349, −1.89643808709161417825075195642, −1.25142272857421120642537715495,
0.55785310160123146073860127197, 2.00317303001226535814907685180, 2.26438631425944467530577621511, 4.01414916361127140779733572671, 4.43983322219929023445699041637, 5.40860695669416206599223194878, 6.30870240184479781414346455508, 6.84051038908837821885348764698, 7.79705625149545841750441338906, 8.620107885091210917490345026852
