| L(s) = 1 | − 3.41·2-s − 6.29·3-s + 3.65·4-s − 5·5-s + 21.4·6-s + 9.26·7-s + 14.8·8-s + 12.5·9-s + 17.0·10-s + 11·11-s − 23.0·12-s + 49.0·13-s − 31.6·14-s + 31.4·15-s − 79.8·16-s − 126.·17-s − 43.0·18-s − 19·19-s − 18.2·20-s − 58.2·21-s − 37.5·22-s − 194.·23-s − 93.2·24-s + 25·25-s − 167.·26-s + 90.6·27-s + 33.8·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 1.21·3-s + 0.457·4-s − 0.447·5-s + 1.46·6-s + 0.500·7-s + 0.655·8-s + 0.466·9-s + 0.539·10-s + 0.301·11-s − 0.553·12-s + 1.04·13-s − 0.603·14-s + 0.541·15-s − 1.24·16-s − 1.80·17-s − 0.563·18-s − 0.229·19-s − 0.204·20-s − 0.605·21-s − 0.363·22-s − 1.75·23-s − 0.793·24-s + 0.200·25-s − 1.26·26-s + 0.646·27-s + 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 3.41T + 8T^{2} \) |
| 3 | \( 1 + 6.29T + 27T^{2} \) |
| 7 | \( 1 - 9.26T + 343T^{2} \) |
| 13 | \( 1 - 49.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 126.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 64.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 28.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 272.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 204.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 46.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 311.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 74.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 398.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 17.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 212.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 374.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 31.0T + 9.12e5T^{2} \) |
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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
−8.985182606314951302885596576018, −8.434070316946658367462064873238, −7.63125180342232555126088668156, −6.58296592900971713247913159534, −5.98167386494679419620182066428, −4.67207758834016765539996409939, −4.08258800293000381458674890186, −2.11665235612465352955051143207, −0.925545776420434376882952696764, 0,
0.925545776420434376882952696764, 2.11665235612465352955051143207, 4.08258800293000381458674890186, 4.67207758834016765539996409939, 5.98167386494679419620182066428, 6.58296592900971713247913159534, 7.63125180342232555126088668156, 8.434070316946658367462064873238, 8.985182606314951302885596576018
