L(s) = 1 | + 7.58i·2-s − 5.19i·3-s − 41.4·4-s − 47.0i·5-s + 39.3·6-s + 16.7i·7-s − 193. i·8-s − 27·9-s + 356.·10-s + 0.510i·11-s + 215. i·12-s + 290. i·13-s − 126.·14-s − 244.·15-s + 801.·16-s − 349.·17-s + ⋯ |
L(s) = 1 | + 1.89i·2-s − 0.577i·3-s − 2.59·4-s − 1.88i·5-s + 1.09·6-s + 0.341i·7-s − 3.01i·8-s − 0.333·9-s + 3.56·10-s + 0.00421i·11-s + 1.49i·12-s + 1.71i·13-s − 0.646·14-s − 1.08·15-s + 3.13·16-s − 1.21·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00587i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7494493670\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7494493670\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19iT \) |
| 67 | \( 1 + (-26.3 + 4.48e3i)T \) |
good | 2 | \( 1 - 7.58iT - 16T^{2} \) |
| 5 | \( 1 + 47.0iT - 625T^{2} \) |
| 7 | \( 1 - 16.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 0.510iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 290. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 349.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 108.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 481.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 509.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.34e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 176.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.76e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.92e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 183.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 386. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.49e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 650. iT - 1.38e7T^{2} \) |
| 71 | \( 1 + 8.62e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 7.85e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.67e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.40e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.11e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 9.40e3iT - 8.85e7T^{2} \) |
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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
−12.71548895154086740527193295037, −11.72384812710455031890307022156, −9.421833629190173782079669606890, −8.885969836110138526644496125416, −8.301331639747642663422196338839, −7.06848172920591443844200673982, −6.19414788298150648140321420522, −4.98596071676073303640885690726, −4.42655852674055885968776249922, −1.25981890164029919574814130687,
0.29727640147498862251485821515, 2.43279165186858093608385829704, 3.15849683818273178765433973133, 4.11915337633351269770462525174, 5.64663042859958440195665727983, 7.35020282552464152352489580942, 8.749179176022593547261742925635, 9.988839020507093245330355703801, 10.52947333182947358132387282291, 10.98506422955515318370187703939