| L(s) = 1 | − 3.48·2-s − 0.850·3-s + 4.14·4-s + 2.96·6-s − 7·7-s + 13.4·8-s − 26.2·9-s − 6.90·11-s − 3.52·12-s + 22.1·13-s + 24.3·14-s − 79.9·16-s − 88.3·17-s + 91.5·18-s + 36.9·19-s + 5.95·21-s + 24.0·22-s + 95.5·23-s − 11.4·24-s − 77.1·26-s + 45.2·27-s − 29.0·28-s + 269.·29-s + 197.·31-s + 171.·32-s + 5.87·33-s + 307.·34-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.163·3-s + 0.518·4-s + 0.201·6-s − 0.377·7-s + 0.593·8-s − 0.973·9-s − 0.189·11-s − 0.0848·12-s + 0.472·13-s + 0.465·14-s − 1.24·16-s − 1.25·17-s + 1.19·18-s + 0.446·19-s + 0.0618·21-s + 0.233·22-s + 0.866·23-s − 0.0970·24-s − 0.582·26-s + 0.322·27-s − 0.196·28-s + 1.72·29-s + 1.14·31-s + 0.946·32-s + 0.0309·33-s + 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6230875634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6230875634\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 3.48T + 8T^{2} \) |
| 3 | \( 1 + 0.850T + 27T^{2} \) |
| 11 | \( 1 + 6.90T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 88.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 95.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 2.14T + 5.06e4T^{2} \) |
| 41 | \( 1 - 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 17.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 528.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 641.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 142.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 478.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 105.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 986.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 711.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 636.T + 9.12e5T^{2} \) |
| 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
−11.93883149093065659701927061456, −10.97290372419542313598210532952, −10.22546506616105285287166021515, −8.991951792878873405236081482564, −8.531341396647836814886256443458, −7.25052206559731226581520476539, −6.12239047875226976596484513044, −4.59256594390212593096079105703, −2.69644973675926636942754965773, −0.74630730814469119042364174987,
0.74630730814469119042364174987, 2.69644973675926636942754965773, 4.59256594390212593096079105703, 6.12239047875226976596484513044, 7.25052206559731226581520476539, 8.531341396647836814886256443458, 8.991951792878873405236081482564, 10.22546506616105285287166021515, 10.97290372419542313598210532952, 11.93883149093065659701927061456
