L(s) = 1 | + 0.810·2-s − 1.66·3-s − 1.34·4-s − 1.34·6-s + 5.21·7-s − 2.70·8-s − 0.243·9-s + 3.14·11-s + 2.22·12-s − 5.93·13-s + 4.22·14-s + 0.488·16-s − 3.59·17-s − 0.197·18-s − 8.65·21-s + 2.54·22-s + 8.45·23-s + 4.49·24-s − 4.81·26-s + 5.38·27-s − 6.99·28-s − 6.84·29-s − 3.11·31-s + 5.81·32-s − 5.21·33-s − 2.91·34-s + 0.326·36-s + ⋯ |
L(s) = 1 | + 0.573·2-s − 0.958·3-s − 0.671·4-s − 0.549·6-s + 1.97·7-s − 0.958·8-s − 0.0811·9-s + 0.947·11-s + 0.643·12-s − 1.64·13-s + 1.12·14-s + 0.122·16-s − 0.871·17-s − 0.0464·18-s − 1.88·21-s + 0.542·22-s + 1.76·23-s + 0.918·24-s − 0.943·26-s + 1.03·27-s − 1.32·28-s − 1.27·29-s − 0.560·31-s + 1.02·32-s − 0.907·33-s − 0.499·34-s + 0.0544·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.493680381\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.493680381\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 0.810T + 2T^{2} \) |
| 3 | \( 1 + 1.66T + 3T^{2} \) |
| 7 | \( 1 - 5.21T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 + 5.93T + 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 23 | \( 1 - 8.45T + 23T^{2} \) |
| 29 | \( 1 + 6.84T + 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 + 0.373T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 - 1.04T + 47T^{2} \) |
| 53 | \( 1 + 2.81T + 53T^{2} \) |
| 59 | \( 1 + 7.22T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 + 0.749T + 67T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 + 7.36T + 73T^{2} \) |
| 79 | \( 1 - 9.10T + 79T^{2} \) |
| 83 | \( 1 - 9.47T + 83T^{2} \) |
| 89 | \( 1 - 9.87T + 89T^{2} \) |
| 97 | \( 1 - 5.88T + 97T^{2} \) |
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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
−7.58454421651071492478877336328, −7.03521508876979913475084208989, −6.14664897831691633897072506418, −5.35340464193596836896934241219, −4.90369134306048134674494806543, −4.65284731247750177947485315184, −3.77040153292991172753044314241, −2.62721673676774030932472740319, −1.65996224025344048189298552825, −0.58517092705940007024128709237,
0.58517092705940007024128709237, 1.65996224025344048189298552825, 2.62721673676774030932472740319, 3.77040153292991172753044314241, 4.65284731247750177947485315184, 4.90369134306048134674494806543, 5.35340464193596836896934241219, 6.14664897831691633897072506418, 7.03521508876979913475084208989, 7.58454421651071492478877336328