| L(s) = 1 | + (−0.415 + 0.909i)3-s + (0.920 − 1.06i)5-s + (−3.86 + 2.48i)7-s + (−0.654 − 0.755i)9-s + (−0.178 + 1.24i)11-s + (−3.62 − 2.32i)13-s + (0.583 + 1.27i)15-s + (−3.37 − 0.989i)17-s + (−0.828 + 0.243i)19-s + (−0.653 − 4.54i)21-s + (−2.01 + 4.35i)23-s + (0.430 + 2.99i)25-s + (0.959 − 0.281i)27-s + (−0.901 − 0.264i)29-s + (−3.62 − 7.92i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 + 0.525i)3-s + (0.411 − 0.475i)5-s + (−1.45 + 0.937i)7-s + (−0.218 − 0.251i)9-s + (−0.0538 + 0.374i)11-s + (−1.00 − 0.645i)13-s + (0.150 + 0.330i)15-s + (−0.817 − 0.240i)17-s + (−0.189 + 0.0557i)19-s + (−0.142 − 0.991i)21-s + (−0.420 + 0.907i)23-s + (0.0860 + 0.598i)25-s + (0.184 − 0.0542i)27-s + (−0.167 − 0.0491i)29-s + (−0.650 − 1.42i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00594776 + 0.283578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00594776 + 0.283578i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (2.01 - 4.35i)T \) |
| good | 5 | \( 1 + (-0.920 + 1.06i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (3.86 - 2.48i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.178 - 1.24i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.62 + 2.32i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (3.37 + 0.989i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.828 - 0.243i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.901 + 0.264i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (3.62 + 7.92i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.07 - 2.39i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (5.66 - 6.53i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.334 + 0.731i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 0.337T + 47T^{2} \) |
| 53 | \( 1 + (5.96 - 3.83i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (4.99 + 3.20i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (2.75 + 6.04i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.63 - 11.3i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.21 - 8.44i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-14.6 + 4.29i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (9.38 + 6.03i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.65 - 4.21i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.27 + 9.36i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (2.32 - 2.67i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.24646673662367408513658120504, −9.945935913406554737134286141058, −9.612978030045208749193741802822, −8.936879647360625467308417966737, −7.64498760705780647985385087148, −6.45413812122709850915471328324, −5.66830445523860094898490328178, −4.82390180669344221909284391709, −3.43745700780943083773371437315, −2.31289621361414061106573104593,
0.15234234263473587655730084066, 2.19946567131178557096441688674, 3.38616418308489564607579257647, 4.63270281270223155562553016392, 6.10778671668642273320946844305, 6.71515257496061263242657289402, 7.28586224835084996873394033478, 8.638489779448328765535874405717, 9.630060091977206333236926397787, 10.39445152584880827533814164466
