L(s) = 1 | + (0.0475 + 0.998i)2-s + (1.50 − 0.442i)3-s + (−0.995 + 0.0950i)4-s + (0.514 + 1.48i)6-s + (−0.142 − 0.989i)8-s + (1.23 − 0.795i)9-s + (0.0930 − 0.268i)11-s + (−1.45 + 0.584i)12-s + (0.981 − 0.189i)16-s + (−1.15 − 0.110i)17-s + (0.853 + 1.19i)18-s + (−1.74 + 0.899i)19-s + (0.273 + 0.0801i)22-s + (−0.653 − 1.43i)24-s + (−0.142 + 0.989i)25-s + ⋯ |
L(s) = 1 | + (0.0475 + 0.998i)2-s + (1.50 − 0.442i)3-s + (−0.995 + 0.0950i)4-s + (0.514 + 1.48i)6-s + (−0.142 − 0.989i)8-s + (1.23 − 0.795i)9-s + (0.0930 − 0.268i)11-s + (−1.45 + 0.584i)12-s + (0.981 − 0.189i)16-s + (−1.15 − 0.110i)17-s + (0.853 + 1.19i)18-s + (−1.74 + 0.899i)19-s + (0.273 + 0.0801i)22-s + (−0.653 − 1.43i)24-s + (−0.142 + 0.989i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.246629414\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246629414\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0475 - 0.998i)T \) |
| 67 | \( 1 + (0.654 + 0.755i)T \) |
good | 3 | \( 1 + (-1.50 + 0.442i)T + (0.841 - 0.540i)T^{2} \) |
| 5 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 7 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 11 | \( 1 + (-0.0930 + 0.268i)T + (-0.786 - 0.618i)T^{2} \) |
| 13 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 17 | \( 1 + (1.15 + 0.110i)T + (0.981 + 0.189i)T^{2} \) |
| 19 | \( 1 + (1.74 - 0.899i)T + (0.580 - 0.814i)T^{2} \) |
| 23 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.975 + 1.37i)T + (-0.327 - 0.945i)T^{2} \) |
| 43 | \( 1 + (0.271 + 0.595i)T + (-0.654 + 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 71 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 73 | \( 1 + (-0.651 - 1.88i)T + (-0.786 + 0.618i)T^{2} \) |
| 79 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 83 | \( 1 + (1.28 - 0.247i)T + (0.928 - 0.371i)T^{2} \) |
| 89 | \( 1 + (0.0913 + 0.0268i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−11.01282271909130652820004300465, −9.853767924435592699006593495310, −8.877121475221182621244684216523, −8.558990329620347339534781507375, −7.62388643859935597021185150689, −6.85939645007146186788041314809, −5.87272733978529516498529031658, −4.38357172718491100256091532316, −3.51877897408444117894329912764, −2.06538464953225577092620536470,
2.10281290068202996215980681425, 2.81525805335228881672199546626, 4.13535074264435909667938881766, 4.58228298823544602731557090694, 6.37059521103547196146704955349, 7.79035125214781339254220288253, 8.727710517442326891649202626098, 9.036856589761190851032401412429, 10.06323513059919787522288241628, 10.71852648214121148609895814497