L(s) = 1 | − 2.46·2-s − 3-s + 4.07·4-s − 5-s + 2.46·6-s − 4.85·7-s − 5.11·8-s + 9-s + 2.46·10-s + 3.32·11-s − 4.07·12-s − 13-s + 11.9·14-s + 15-s + 4.44·16-s − 6.96·17-s − 2.46·18-s − 1.90·19-s − 4.07·20-s + 4.85·21-s − 8.20·22-s − 0.550·23-s + 5.11·24-s + 25-s + 2.46·26-s − 27-s − 19.7·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s − 0.577·3-s + 2.03·4-s − 0.447·5-s + 1.00·6-s − 1.83·7-s − 1.80·8-s + 0.333·9-s + 0.779·10-s + 1.00·11-s − 1.17·12-s − 0.277·13-s + 3.19·14-s + 0.258·15-s + 1.11·16-s − 1.69·17-s − 0.580·18-s − 0.436·19-s − 0.910·20-s + 1.05·21-s − 1.74·22-s − 0.114·23-s + 1.04·24-s + 0.200·25-s + 0.483·26-s − 0.192·27-s − 3.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0009574605259\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0009574605259\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 17 | \( 1 + 6.96T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 + 0.550T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 37 | \( 1 + 3.69T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 1.42T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 0.524T + 59T^{2} \) |
| 61 | \( 1 - 8.37T + 61T^{2} \) |
| 67 | \( 1 + 2.34T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 3.51T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 + 8.64T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.391808390166076577478790547146, −7.06229441506679215773117680668, −6.90895153654305781963671827926, −6.48102841752895301900440151712, −5.55669106398932837939549028315, −4.24422419216925663377954650162, −3.51542077908264703831592410413, −2.47404501516344772802851966601, −1.50035232396133220811199233103, −0.02174827262580131625005937040,
0.02174827262580131625005937040, 1.50035232396133220811199233103, 2.47404501516344772802851966601, 3.51542077908264703831592410413, 4.24422419216925663377954650162, 5.55669106398932837939549028315, 6.48102841752895301900440151712, 6.90895153654305781963671827926, 7.06229441506679215773117680668, 8.391808390166076577478790547146