| L(s) = 1 | − 3.22·3-s − 1.96·5-s + 3.83·7-s + 7.39·9-s + 11-s − 4.08·13-s + 6.31·15-s + 6.84·17-s − 19-s − 12.3·21-s − 6.67·23-s − 1.15·25-s − 14.1·27-s + 3.43·29-s − 5.74·31-s − 3.22·33-s − 7.52·35-s + 4.55·37-s + 13.1·39-s − 0.970·41-s − 6.42·43-s − 14.4·45-s + 7.92·47-s + 7.71·49-s − 22.0·51-s + 13.4·53-s − 1.96·55-s + ⋯ |
| L(s) = 1 | − 1.86·3-s − 0.876·5-s + 1.45·7-s + 2.46·9-s + 0.301·11-s − 1.13·13-s + 1.63·15-s + 1.65·17-s − 0.229·19-s − 2.69·21-s − 1.39·23-s − 0.231·25-s − 2.72·27-s + 0.638·29-s − 1.03·31-s − 0.561·33-s − 1.27·35-s + 0.748·37-s + 2.10·39-s − 0.151·41-s − 0.980·43-s − 2.15·45-s + 1.15·47-s + 1.10·49-s − 3.08·51-s + 1.84·53-s − 0.264·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7835255032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7835255032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.22T + 3T^{2} \) |
| 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 + 5.74T + 31T^{2} \) |
| 37 | \( 1 - 4.55T + 37T^{2} \) |
| 41 | \( 1 + 0.970T + 41T^{2} \) |
| 43 | \( 1 + 6.42T + 43T^{2} \) |
| 47 | \( 1 - 7.92T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 9.76T + 61T^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 + 7.97T + 71T^{2} \) |
| 73 | \( 1 - 4.01T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 7.57T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−9.714299615869186467047231726759, −8.238063015382679056477016872092, −7.63008240148966295308437792951, −7.04942114971721527046441581451, −5.87503763589225241672356387931, −5.30122767841790200575234400225, −4.55076569108602696311151456679, −3.85924631497266751733531309786, −1.88229548735201868574724885503, −0.69115957679914646428669007383,
0.69115957679914646428669007383, 1.88229548735201868574724885503, 3.85924631497266751733531309786, 4.55076569108602696311151456679, 5.30122767841790200575234400225, 5.87503763589225241672356387931, 7.04942114971721527046441581451, 7.63008240148966295308437792951, 8.238063015382679056477016872092, 9.714299615869186467047231726759
