| L(s) = 1 | − 0.200·2-s + 2.02·3-s − 1.95·4-s + 1.96·5-s − 0.405·6-s − 2.70·7-s + 0.795·8-s + 1.08·9-s − 0.394·10-s − 4.18·11-s − 3.95·12-s + 1.98·13-s + 0.543·14-s + 3.97·15-s + 3.75·16-s + 3.68·17-s − 0.217·18-s − 6.87·19-s − 3.85·20-s − 5.46·21-s + 0.840·22-s − 7.32·23-s + 1.60·24-s − 1.13·25-s − 0.397·26-s − 3.87·27-s + 5.29·28-s + ⋯ |
| L(s) = 1 | − 0.141·2-s + 1.16·3-s − 0.979·4-s + 0.879·5-s − 0.165·6-s − 1.02·7-s + 0.281·8-s + 0.360·9-s − 0.124·10-s − 1.26·11-s − 1.14·12-s + 0.549·13-s + 0.145·14-s + 1.02·15-s + 0.939·16-s + 0.893·17-s − 0.0511·18-s − 1.57·19-s − 0.861·20-s − 1.19·21-s + 0.179·22-s − 1.52·23-s + 0.327·24-s − 0.226·25-s − 0.0780·26-s − 0.746·27-s + 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.200T + 2T^{2} \) |
| 3 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 - 1.96T + 5T^{2} \) |
| 7 | \( 1 + 2.70T + 7T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 + 6.87T + 19T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 - 1.13T + 29T^{2} \) |
| 31 | \( 1 - 8.28T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 47 | \( 1 - 6.62T + 47T^{2} \) |
| 53 | \( 1 + 4.56T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 9.53T + 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 - 5.94T + 71T^{2} \) |
| 73 | \( 1 - 3.82T + 73T^{2} \) |
| 79 | \( 1 + 8.12T + 79T^{2} \) |
| 83 | \( 1 + 5.00T + 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 97 | \( 1 - 3.54T + 97T^{2} \) |
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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
−8.777109069279277081652325150474, −8.280290638865445907378788056813, −7.66367899459499867499733708274, −6.25182095875489524406140715956, −5.78015969988275956637492248009, −4.61911944210037655904666451348, −3.62394780134905670980545763660, −2.86806041356263242939344921720, −1.86504676172559187932549600451, 0,
1.86504676172559187932549600451, 2.86806041356263242939344921720, 3.62394780134905670980545763660, 4.61911944210037655904666451348, 5.78015969988275956637492248009, 6.25182095875489524406140715956, 7.66367899459499867499733708274, 8.280290638865445907378788056813, 8.777109069279277081652325150474
