L(s) = 1 | − 2-s + 4-s − 1.80·7-s − 8-s + 9-s − 1.24·11-s + 1.80·14-s + 16-s − 0.445·17-s − 18-s + 0.445·19-s + 1.24·22-s + 1.24·23-s + 25-s − 1.80·28-s + 0.445·29-s − 0.445·31-s − 32-s + 0.445·34-s + 36-s + 1.80·37-s − 0.445·38-s + 1.24·41-s + 1.80·43-s − 1.24·44-s − 1.24·46-s + 2.24·49-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 1.80·7-s − 8-s + 9-s − 1.24·11-s + 1.80·14-s + 16-s − 0.445·17-s − 18-s + 0.445·19-s + 1.24·22-s + 1.24·23-s + 25-s − 1.80·28-s + 0.445·29-s − 0.445·31-s − 32-s + 0.445·34-s + 36-s + 1.80·37-s − 0.445·38-s + 1.24·41-s + 1.80·43-s − 1.24·44-s − 1.24·46-s + 2.24·49-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6040575604\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6040575604\) |
\(L(1)\) |
|
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 | \( 1 + 1.80T + T^{2} \) |
| 11 | \( 1 + 1.24T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 - 0.445T + T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - 0.445T + T^{2} \) |
| 31 | \( 1 + 0.445T + T^{2} \) |
| 37 | \( 1 - 1.80T + T^{2} \) |
| 41 | \( 1 - 1.24T + T^{2} \) |
| 43 | \( 1 - 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.24T + T^{2} \) |
| 59 | \( 1 - 1.80T + T^{2} \) |
| 61 | \( 1 + 1.24T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.80T + T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.80T + T^{2} \) |
| 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
−9.412980528181141553684774170636, −8.803388753055163419314371691255, −7.60770011685702659017541070939, −7.19308054045805578545423018584, −6.41737218706723170672936707924, −5.65734402908629410094255396023, −4.37964919113387758996924017437, −3.07884610974690439801936395900, −2.58292251964824413891401667894, −0.875377605188674701690736104607,
0.875377605188674701690736104607, 2.58292251964824413891401667894, 3.07884610974690439801936395900, 4.37964919113387758996924017437, 5.65734402908629410094255396023, 6.41737218706723170672936707924, 7.19308054045805578545423018584, 7.60770011685702659017541070939, 8.803388753055163419314371691255, 9.412980528181141553684774170636