| L(s) = 1 | − 2-s + 3-s + 4-s + 3.05·5-s − 6-s + 1.53·7-s − 8-s + 9-s − 3.05·10-s − 3.38·11-s + 12-s − 1.02·13-s − 1.53·14-s + 3.05·15-s + 16-s − 17-s − 18-s + 3.86·19-s + 3.05·20-s + 1.53·21-s + 3.38·22-s + 3.31·23-s − 24-s + 4.34·25-s + 1.02·26-s + 27-s + 1.53·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.36·5-s − 0.408·6-s + 0.580·7-s − 0.353·8-s + 0.333·9-s − 0.966·10-s − 1.01·11-s + 0.288·12-s − 0.283·13-s − 0.410·14-s + 0.789·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.885·19-s + 0.683·20-s + 0.335·21-s + 0.721·22-s + 0.691·23-s − 0.204·24-s + 0.869·25-s + 0.200·26-s + 0.192·27-s + 0.290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582476612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582476612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 3.05T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 19 | \( 1 - 3.86T + 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 - 9.99T + 29T^{2} \) |
| 31 | \( 1 + 0.616T + 31T^{2} \) |
| 37 | \( 1 - 6.96T + 37T^{2} \) |
| 41 | \( 1 - 3.75T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 + 4.62T + 53T^{2} \) |
| 61 | \( 1 + 7.29T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 3.33T + 71T^{2} \) |
| 73 | \( 1 + 6.75T + 73T^{2} \) |
| 79 | \( 1 - 9.75T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 6.45T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−8.108680768078744156161699776754, −7.60209715949024757195496855614, −6.72179416306849607011098980578, −6.09146952357423584217797024050, −5.14452013025369148099512385013, −4.73272109022745040524684943752, −3.21893926815618757636543110178, −2.57684737901484364137224665864, −1.88422329340514350656408090316, −0.946402405965776991617956002487,
0.946402405965776991617956002487, 1.88422329340514350656408090316, 2.57684737901484364137224665864, 3.21893926815618757636543110178, 4.73272109022745040524684943752, 5.14452013025369148099512385013, 6.09146952357423584217797024050, 6.72179416306849607011098980578, 7.60209715949024757195496855614, 8.108680768078744156161699776754
