| L(s) = 1 | + 13.0·2-s − 81·3-s − 342.·4-s − 5.38·5-s − 1.05e3·6-s − 6.26e3·7-s − 1.11e4·8-s + 6.56e3·9-s − 69.9·10-s − 4.36e4·11-s + 2.77e4·12-s − 6.91e4·13-s − 8.14e4·14-s + 436.·15-s + 3.11e4·16-s − 3.25e5·17-s + 8.52e4·18-s − 6.28e5·19-s + 1.84e3·20-s + 5.07e5·21-s − 5.66e5·22-s − 4.08e5·23-s + 9.00e5·24-s − 1.95e6·25-s − 8.98e5·26-s − 5.31e5·27-s + 2.14e6·28-s + ⋯ |
| L(s) = 1 | + 0.574·2-s − 0.577·3-s − 0.669·4-s − 0.00385·5-s − 0.331·6-s − 0.985·7-s − 0.959·8-s + 0.333·9-s − 0.00221·10-s − 0.898·11-s + 0.386·12-s − 0.671·13-s − 0.566·14-s + 0.00222·15-s + 0.118·16-s − 0.945·17-s + 0.191·18-s − 1.10·19-s + 0.00258·20-s + 0.569·21-s − 0.516·22-s − 0.304·23-s + 0.553·24-s − 0.999·25-s − 0.385·26-s − 0.192·27-s + 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.01551290879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01551290879\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 13.0T + 512T^{2} \) |
| 5 | \( 1 + 5.38T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.26e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.36e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.91e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.25e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 4.08e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.39e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.85e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.07e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.00e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.29e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.79e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.75e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.32e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.56e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.57e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.62e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.14e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.93e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.84e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.61e8T + 7.60e17T^{2} \) |
| 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
−11.08187483002978631751960095091, −9.971122796864254818503355280958, −9.221417924328501794120117486953, −7.894164810721956530830122131257, −6.54654579803220586211712714093, −5.66963008600234600939654774186, −4.64105490836701662928814818262, −3.62244763503560949269388741171, −2.25557841807134978345707101436, −0.05402096607047457878237354645,
0.05402096607047457878237354645, 2.25557841807134978345707101436, 3.62244763503560949269388741171, 4.64105490836701662928814818262, 5.66963008600234600939654774186, 6.54654579803220586211712714093, 7.894164810721956530830122131257, 9.221417924328501794120117486953, 9.971122796864254818503355280958, 11.08187483002978631751960095091
