| L(s) = 1 | + (−0.821 − 1.15i)2-s + (−0.649 + 1.89i)4-s + (−0.426 − 0.155i)5-s + (−1.28 − 0.225i)7-s + (2.71 − 0.807i)8-s + (0.171 + 0.618i)10-s + (−0.00602 − 0.0165i)11-s + (−1.43 − 1.71i)13-s + (0.793 + 1.66i)14-s + (−3.15 − 2.45i)16-s + (−1.27 + 0.734i)17-s + (0.677 − 1.17i)19-s + (0.570 − 0.706i)20-s + (−0.0141 + 0.0205i)22-s + (0.369 + 2.09i)23-s + ⋯ |
| L(s) = 1 | + (−0.581 − 0.813i)2-s + (−0.324 + 0.945i)4-s + (−0.190 − 0.0694i)5-s + (−0.484 − 0.0854i)7-s + (0.958 − 0.285i)8-s + (0.0543 + 0.195i)10-s + (−0.00181 − 0.00499i)11-s + (−0.398 − 0.474i)13-s + (0.211 + 0.443i)14-s + (−0.789 − 0.614i)16-s + (−0.308 + 0.178i)17-s + (0.155 − 0.269i)19-s + (0.127 − 0.157i)20-s + (−0.00300 + 0.00438i)22-s + (0.0770 + 0.436i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0431591 + 0.220945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0431591 + 0.220945i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.821 + 1.15i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.426 + 0.155i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.28 + 0.225i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.00602 + 0.0165i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.43 + 1.71i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.27 - 0.734i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.677 + 1.17i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.369 - 2.09i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.56 + 4.67i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (8.87 - 1.56i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.58 + 2.64i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.56 + 1.86i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (10.1 - 3.71i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.791 - 4.48i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + (-3.75 + 10.3i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-8.87 - 1.56i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.70 + 3.10i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.03 - 8.72i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.339 - 0.587i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.19 - 6.19i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.29 - 1.53i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.103 - 0.0598i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.4 - 3.81i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.832554364287729428941393400130, −9.565448823340626005266427848707, −8.384593529898591589758834437979, −7.68058861160913786421953377659, −6.71720273751858594130358366033, −5.36607950167481531218978391058, −4.11834774865644943025096292304, −3.20724492089420906939908849369, −1.94193838813524865304799651817, −0.14122817734785122694728377188,
1.85514482696932029548060522313, 3.59055625196653764676712384228, 4.86012338773085891820238322365, 5.79400213629901599982106264354, 6.79514114818275542489298540075, 7.43267990656346243104374558880, 8.420614419452046113953353021714, 9.321783673161795370532513742069, 9.853746430645175745278668437406, 10.91273112882465714775707018324
