| L(s) = 1 | + (1.35 + 0.403i)2-s + (1.25 − 0.857i)3-s + (1.67 + 1.09i)4-s + (−0.263 + 0.0197i)5-s + (2.05 − 0.654i)6-s + (−2.03 − 1.68i)7-s + (1.82 + 2.15i)8-s + (−0.248 + 0.633i)9-s + (−0.365 − 0.0796i)10-s + (−1.57 + 0.619i)11-s + (3.04 − 0.0592i)12-s + (−2.83 − 2.26i)13-s + (−2.08 − 3.11i)14-s + (−0.314 + 0.250i)15-s + (1.60 + 3.66i)16-s + (0.766 − 0.826i)17-s + ⋯ |
| L(s) = 1 | + (0.958 + 0.285i)2-s + (0.726 − 0.495i)3-s + (0.837 + 0.547i)4-s + (−0.117 + 0.00883i)5-s + (0.837 − 0.267i)6-s + (−0.770 − 0.637i)7-s + (0.646 + 0.763i)8-s + (−0.0828 + 0.211i)9-s + (−0.115 − 0.0251i)10-s + (−0.475 + 0.186i)11-s + (0.879 − 0.0171i)12-s + (−0.787 − 0.628i)13-s + (−0.555 − 0.831i)14-s + (−0.0812 + 0.0647i)15-s + (0.401 + 0.915i)16-s + (0.186 − 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.20582 + 0.0865345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20582 + 0.0865345i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 - 0.403i)T \) |
| 7 | \( 1 + (2.03 + 1.68i)T \) |
| good | 3 | \( 1 + (-1.25 + 0.857i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.263 - 0.0197i)T + (4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (1.57 - 0.619i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (2.83 + 2.26i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.826i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.734 - 1.27i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.16 - 2.33i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.25 + 5.50i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.89 + 5.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.55 - 0.788i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-2.59 + 5.39i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.36 - 9.06i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (5.64 + 0.851i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (0.0663 + 0.0204i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.838 + 11.1i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.0587 - 0.190i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (12.8 + 7.43i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.45 + 0.788i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.419 - 2.78i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-13.0 + 7.52i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.20 - 4.01i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-16.7 - 6.56i)T + (65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 7.85iT - 97T^{2} \) |
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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
−12.89547044403832068567333334591, −11.87878660792700989403179406672, −10.66809281546744193528912280947, −9.580057865297333426558080668902, −7.80207197757382840086971476626, −7.63336062646691950924405258980, −6.27781017619137632199314838420, −4.98552219799263926776355298090, −3.53016559047383233159987784731, −2.43631675045396150247809288484,
2.54556635541983140544861144260, 3.47215778009560177206858952066, 4.77222104611312650338187215344, 6.03312435458143452602974934261, 7.15272262454856046664777482421, 8.686503280709991616814536430390, 9.634964756716799669623887238506, 10.50154675760623991073548852579, 11.85099571598472841018628221679, 12.44099446262554630126425127532
