| L(s) = 1 | + 2.83·2-s + 8.17·3-s + 0.0460·4-s + 2.30·5-s + 23.1·6-s − 3.15·7-s − 22.5·8-s + 39.8·9-s + 6.55·10-s + 11·11-s + 0.376·12-s − 8.95·14-s + 18.8·15-s − 64.3·16-s − 127.·17-s + 113.·18-s − 108.·19-s + 0.106·20-s − 25.8·21-s + 31.2·22-s − 19.0·23-s − 184.·24-s − 119.·25-s + 105.·27-s − 0.145·28-s + 197.·29-s + 53.5·30-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 1.57·3-s + 0.00576·4-s + 0.206·5-s + 1.57·6-s − 0.170·7-s − 0.997·8-s + 1.47·9-s + 0.207·10-s + 0.301·11-s + 0.00906·12-s − 0.171·14-s + 0.325·15-s − 1.00·16-s − 1.82·17-s + 1.48·18-s − 1.30·19-s + 0.00118·20-s − 0.268·21-s + 0.302·22-s − 0.172·23-s − 1.56·24-s − 0.957·25-s + 0.749·27-s − 0.000982·28-s + 1.26·29-s + 0.325·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.83T + 8T^{2} \) |
| 3 | \( 1 - 8.17T + 27T^{2} \) |
| 5 | \( 1 - 2.30T + 125T^{2} \) |
| 7 | \( 1 + 3.15T + 343T^{2} \) |
| 17 | \( 1 + 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 197.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 326.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 406.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 625.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 399.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 583.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 578.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 103.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 660.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−8.548402526785546302902160436654, −7.904413423515924765204977501186, −6.66908698814175764846004959883, −6.21248937719520089927614631621, −4.87175502418792701417322283606, −4.17779769712151695091107799151, −3.55508894297545839387384886120, −2.56306228738241455283689161653, −1.93082469657540663396342550115, 0,
1.93082469657540663396342550115, 2.56306228738241455283689161653, 3.55508894297545839387384886120, 4.17779769712151695091107799151, 4.87175502418792701417322283606, 6.21248937719520089927614631621, 6.66908698814175764846004959883, 7.904413423515924765204977501186, 8.548402526785546302902160436654
