| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 17.2·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s − 34.4·10-s − 32.9·11-s − 12·12-s + 53.2·13-s − 14·14-s − 51.7·15-s + 16·16-s + 9.19·17-s − 18·18-s + 19·19-s + 68.9·20-s − 21·21-s + 65.8·22-s + 84.8·23-s + 24·24-s + 172.·25-s − 106.·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.54·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.09·10-s − 0.902·11-s − 0.288·12-s + 1.13·13-s − 0.267·14-s − 0.890·15-s + 0.250·16-s + 0.131·17-s − 0.235·18-s + 0.229·19-s + 0.770·20-s − 0.218·21-s + 0.638·22-s + 0.769·23-s + 0.204·24-s + 1.37·25-s − 0.803·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.811343172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.811343172\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 - 17.2T + 125T^{2} \) |
| 11 | \( 1 + 32.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.19T + 4.91e3T^{2} \) |
| 23 | \( 1 - 84.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 115.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 399.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 385.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 174.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 304.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 178.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 625.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 981.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.18e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 894.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 197.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 8.42T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.21e3T + 9.12e5T^{2} \) |
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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
−10.00982730167867242692608749798, −9.099693311518512770368393599515, −8.349445077112739979139121538977, −7.25045346056494797224005915390, −6.27394872549867678360473684666, −5.67763630925191345386451980302, −4.78546687096254945011444800627, −3.02278790808929119845403272800, −1.86315546455017393637257879924, −0.909711074496360599870733505373,
0.909711074496360599870733505373, 1.86315546455017393637257879924, 3.02278790808929119845403272800, 4.78546687096254945011444800627, 5.67763630925191345386451980302, 6.27394872549867678360473684666, 7.25045346056494797224005915390, 8.349445077112739979139121538977, 9.099693311518512770368393599515, 10.00982730167867242692608749798
