L(s) = 1 | − 3.96·2-s − 9·3-s − 16.2·4-s + 45.8·5-s + 35.6·6-s + 214.·7-s + 191.·8-s + 81·9-s − 181.·10-s − 156.·11-s + 146.·12-s + 176.·13-s − 852.·14-s − 412.·15-s − 238.·16-s − 992.·17-s − 321.·18-s − 1.34e3·19-s − 745.·20-s − 1.93e3·21-s + 618.·22-s + 3.04e3·23-s − 1.72e3·24-s − 1.02e3·25-s − 698.·26-s − 729·27-s − 3.49e3·28-s + ⋯ |
L(s) = 1 | − 0.701·2-s − 0.577·3-s − 0.508·4-s + 0.819·5-s + 0.404·6-s + 1.65·7-s + 1.05·8-s + 0.333·9-s − 0.574·10-s − 0.388·11-s + 0.293·12-s + 0.289·13-s − 1.16·14-s − 0.473·15-s − 0.232·16-s − 0.833·17-s − 0.233·18-s − 0.855·19-s − 0.416·20-s − 0.957·21-s + 0.272·22-s + 1.19·23-s − 0.610·24-s − 0.328·25-s − 0.202·26-s − 0.192·27-s − 0.843·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.315071919\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.315071919\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
good | 2 | \( 1 + 3.96T + 32T^{2} \) |
| 5 | \( 1 - 45.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 214.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 156.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 176.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 992.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.34e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.34e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.62e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.59e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.99e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 5.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.52e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.09e4T + 8.58e9T^{2} \) |
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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
−11.41839893623785463219535274166, −10.74911231995618413448661601415, −9.859653841493852940496321391767, −8.669951655138055625222664847813, −8.007616702105701734329570775373, −6.57678091354293285726401611354, −5.15022457436801557254356318819, −4.51657752069108224336575288818, −2.02102192882851117687702967181, −0.884651381840147443018739709513,
0.884651381840147443018739709513, 2.02102192882851117687702967181, 4.51657752069108224336575288818, 5.15022457436801557254356318819, 6.57678091354293285726401611354, 8.007616702105701734329570775373, 8.669951655138055625222664847813, 9.859653841493852940496321391767, 10.74911231995618413448661601415, 11.41839893623785463219535274166