| L(s) = 1 | + 204.·3-s − 258.·5-s + 8.86e3·7-s + 2.21e4·9-s − 3.60e4·11-s − 2.85e4·13-s − 5.29e4·15-s + 3.27e5·17-s − 2.65e5·19-s + 1.81e6·21-s + 2.42e6·23-s − 1.88e6·25-s + 5.09e5·27-s + 3.99e6·29-s + 6.45e6·31-s − 7.38e6·33-s − 2.29e6·35-s − 8.15e6·37-s − 5.84e6·39-s − 7.20e5·41-s + 4.13e7·43-s − 5.74e6·45-s − 3.67e7·47-s + 3.81e7·49-s + 6.69e7·51-s + 1.67e7·53-s + 9.34e6·55-s + ⋯ |
| L(s) = 1 | + 1.45·3-s − 0.185·5-s + 1.39·7-s + 1.12·9-s − 0.743·11-s − 0.277·13-s − 0.270·15-s + 0.950·17-s − 0.467·19-s + 2.03·21-s + 1.80·23-s − 0.965·25-s + 0.184·27-s + 1.04·29-s + 1.25·31-s − 1.08·33-s − 0.258·35-s − 0.715·37-s − 0.404·39-s − 0.0398·41-s + 1.84·43-s − 0.208·45-s − 1.09·47-s + 0.946·49-s + 1.38·51-s + 0.291·53-s + 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.659187376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.659187376\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 3 | \( 1 - 204.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 258.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.86e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.60e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.27e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.42e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.99e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.45e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.15e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.20e5T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.13e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.28e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.91e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.40e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.19e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.44e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.90e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.65e8T + 7.60e17T^{2} \) |
| 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
−10.64334664654949697392773588705, −9.619079419077083341211188302115, −8.434423998729304566430308474066, −8.070923609894968178564335892853, −7.13966175791632708156159976087, −5.33578797336076853350160111911, −4.34635727119769224519790553274, −3.06655046904912396420465479271, −2.19274826127070095583355114948, −1.01077646326770093102294153631,
1.01077646326770093102294153631, 2.19274826127070095583355114948, 3.06655046904912396420465479271, 4.34635727119769224519790553274, 5.33578797336076853350160111911, 7.13966175791632708156159976087, 8.070923609894968178564335892853, 8.434423998729304566430308474066, 9.619079419077083341211188302115, 10.64334664654949697392773588705
