| L(s) = 1 | + (−0.428 + 0.903i)2-s + (−0.103 − 0.359i)3-s + (−0.632 − 0.774i)4-s + (−0.522 − 2.56i)5-s + (0.368 + 0.0599i)6-s + (0.676 − 0.704i)7-s + (0.970 − 0.239i)8-s + (2.41 − 1.52i)9-s + (2.53 + 0.625i)10-s + (3.84 − 1.28i)11-s + (−0.212 + 0.307i)12-s + (−5.19 + 0.844i)13-s + (0.346 + 0.912i)14-s + (−0.865 + 0.454i)15-s + (−0.200 + 0.979i)16-s + (−0.675 + 1.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.303 + 0.638i)2-s + (−0.0600 − 0.207i)3-s + (−0.316 − 0.387i)4-s + (−0.233 − 1.14i)5-s + (0.150 + 0.0244i)6-s + (0.255 − 0.266i)7-s + (0.343 − 0.0846i)8-s + (0.805 − 0.509i)9-s + (0.802 + 0.197i)10-s + (1.15 − 0.387i)11-s + (−0.0612 + 0.0888i)12-s + (−1.44 + 0.234i)13-s + (0.0925 + 0.244i)14-s + (−0.223 + 0.117i)15-s + (−0.0500 + 0.244i)16-s + (−0.163 + 0.432i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.919298 - 0.225713i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.919298 - 0.225713i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.428 - 0.903i)T \) |
| 79 | \( 1 + (-3.98 - 7.94i)T \) |
| good | 3 | \( 1 + (0.103 + 0.359i)T + (-2.53 + 1.60i)T^{2} \) |
| 5 | \( 1 + (0.522 + 2.56i)T + (-4.59 + 1.95i)T^{2} \) |
| 7 | \( 1 + (-0.676 + 0.704i)T + (-0.281 - 6.99i)T^{2} \) |
| 11 | \( 1 + (-3.84 + 1.28i)T + (8.79 - 6.60i)T^{2} \) |
| 13 | \( 1 + (5.19 - 0.844i)T + (12.3 - 4.11i)T^{2} \) |
| 17 | \( 1 + (0.675 - 1.78i)T + (-12.7 - 11.2i)T^{2} \) |
| 19 | \( 1 + (-2.21 - 0.945i)T + (13.1 + 13.7i)T^{2} \) |
| 23 | \( 1 + (-3.03 + 5.26i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.295 - 7.32i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (5.85 - 0.472i)T + (30.5 - 4.97i)T^{2} \) |
| 37 | \( 1 + (7.25 - 5.44i)T + (10.2 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-6.45 + 5.72i)T + (4.94 - 40.7i)T^{2} \) |
| 43 | \( 1 + (0.290 + 0.0970i)T + (34.3 + 25.8i)T^{2} \) |
| 47 | \( 1 + (-5.92 - 4.44i)T + (13.0 + 45.1i)T^{2} \) |
| 53 | \( 1 + (1.15 - 3.98i)T + (-44.7 - 28.3i)T^{2} \) |
| 59 | \( 1 + (4.63 - 5.68i)T + (-11.8 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-0.816 - 6.72i)T + (-59.2 + 14.5i)T^{2} \) |
| 67 | \( 1 + (-5.20 + 7.54i)T + (-23.7 - 62.6i)T^{2} \) |
| 71 | \( 1 + (-12.8 + 3.17i)T + (62.8 - 32.9i)T^{2} \) |
| 73 | \( 1 + (0.0452 + 0.00735i)T + (69.2 + 23.1i)T^{2} \) |
| 83 | \( 1 + (-2.67 - 3.28i)T + (-16.6 + 81.3i)T^{2} \) |
| 89 | \( 1 + (8.14 + 2.00i)T + (78.8 + 41.3i)T^{2} \) |
| 97 | \( 1 + (-1.11 - 9.21i)T + (-94.1 + 23.2i)T^{2} \) |
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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
−12.54613983782499039050182876711, −12.27835651300809649555268410114, −10.71605234291699537814875606254, −9.383056553440072106351490862025, −8.827463584722904574165161673861, −7.51160456272275829337622207176, −6.65437320901097531283281728748, −5.12237936163744711749392486215, −4.12028767880452988791847037845, −1.19733637287341617682799880006,
2.20103790333289811408920935888, 3.70438168507401878588667816617, 5.06798152141297230825966001205, 7.00034293905444024339047616240, 7.58571588644745746193246899411, 9.326878356882586328134017284072, 9.947417995230179863198172750692, 11.06748210296566901981249759035, 11.73067071726918340017769144371, 12.77054589894391725903078731434
