| L(s) = 1 | − 2.56·2-s − 3.68·3-s − 1.43·4-s − 0.561·5-s + 9.43·6-s − 18.1·7-s + 24.1·8-s − 13.4·9-s + 1.43·10-s − 64.7·11-s + 5.30·12-s + 46.5·14-s + 2.06·15-s − 50.4·16-s − 25.5·17-s + 34.3·18-s + 107.·19-s + 0.807·20-s + 66.9·21-s + 165.·22-s + 73.2·23-s − 89.0·24-s − 124.·25-s + 148.·27-s + 26.1·28-s + 175.·29-s − 5.30·30-s + ⋯ |
| L(s) = 1 | − 0.905·2-s − 0.709·3-s − 0.179·4-s − 0.0502·5-s + 0.642·6-s − 0.981·7-s + 1.06·8-s − 0.497·9-s + 0.0454·10-s − 1.77·11-s + 0.127·12-s + 0.888·14-s + 0.0356·15-s − 0.787·16-s − 0.364·17-s + 0.450·18-s + 1.30·19-s + 0.00903·20-s + 0.695·21-s + 1.60·22-s + 0.664·23-s − 0.757·24-s − 0.997·25-s + 1.06·27-s + 0.176·28-s + 1.12·29-s − 0.0322·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3405995093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3405995093\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 2.56T + 8T^{2} \) |
| 3 | \( 1 + 3.68T + 27T^{2} \) |
| 5 | \( 1 + 0.561T + 125T^{2} \) |
| 7 | \( 1 + 18.1T + 343T^{2} \) |
| 11 | \( 1 + 64.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 25.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 69.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 31.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 2.84T + 1.48e5T^{2} \) |
| 59 | \( 1 + 71.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 920.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 541.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 764.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 421.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 583.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.23973082822133045095181957476, −11.00748929899282502274487494874, −10.26324611682658061814236135004, −9.403804637228342808231187292322, −8.264193451209439519268083524647, −7.29060695162877519838169537056, −5.87743755877930293770671208666, −4.83387806920376556936758617363, −2.89880107605525449189059326471, −0.52626949998060217249382417729,
0.52626949998060217249382417729, 2.89880107605525449189059326471, 4.83387806920376556936758617363, 5.87743755877930293770671208666, 7.29060695162877519838169537056, 8.264193451209439519268083524647, 9.403804637228342808231187292322, 10.26324611682658061814236135004, 11.00748929899282502274487494874, 12.23973082822133045095181957476
