L(s) = 1 | − 2-s + 4-s − 0.445·7-s − 8-s + 9-s + 1.80·11-s + 0.445·14-s + 16-s + 1.24·17-s − 18-s − 1.24·19-s − 1.80·22-s − 1.80·23-s + 25-s − 0.445·28-s − 1.24·29-s + 1.24·31-s − 32-s − 1.24·34-s + 36-s + 0.445·37-s + 1.24·38-s − 1.80·41-s + 0.445·43-s + 1.80·44-s + 1.80·46-s − 0.801·49-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 0.445·7-s − 8-s + 9-s + 1.80·11-s + 0.445·14-s + 16-s + 1.24·17-s − 18-s − 1.24·19-s − 1.80·22-s − 1.80·23-s + 25-s − 0.445·28-s − 1.24·29-s + 1.24·31-s − 32-s − 1.24·34-s + 36-s + 0.445·37-s + 1.24·38-s − 1.80·41-s + 0.445·43-s + 1.80·44-s + 1.80·46-s − 0.801·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8571470285\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8571470285\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 0.445T + T^{2} \) |
| 11 | \( 1 - 1.80T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 + 1.24T + T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 + 1.24T + T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 - 0.445T + T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 - 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.80T + T^{2} \) |
| 59 | \( 1 - 0.445T + T^{2} \) |
| 61 | \( 1 - 1.80T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.471545750207664100747207159575, −8.587089118816356606508794573300, −7.943906029771986606418082809963, −6.85032389312096640173458943905, −6.60059441440115932412898267055, −5.66078732644009157287456966480, −4.17007871439038847074469495222, −3.54462451090239104262059073847, −2.11344967430654172826563837789, −1.15651804087806272246957086694,
1.15651804087806272246957086694, 2.11344967430654172826563837789, 3.54462451090239104262059073847, 4.17007871439038847074469495222, 5.66078732644009157287456966480, 6.60059441440115932412898267055, 6.85032389312096640173458943905, 7.943906029771986606418082809963, 8.587089118816356606508794573300, 9.471545750207664100747207159575