| L(s) = 1 | − 3-s + 13-s − 23-s + 25-s + 27-s + 29-s + 31-s − 39-s − 41-s + 47-s + 49-s + 2·59-s + 69-s + 71-s − 73-s − 75-s − 81-s − 87-s − 93-s − 2·101-s + ⋯ |
| L(s) = 1 | − 3-s + 13-s − 23-s + 25-s + 27-s + 29-s + 31-s − 39-s − 41-s + 47-s + 49-s + 2·59-s + 69-s + 71-s − 73-s − 75-s − 81-s − 87-s − 93-s − 2·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7860309256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7860309256\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.941876075251919766856521252555, −8.706928678145890375964307048640, −8.297910188101782088811239332122, −7.03236612956964786597528827910, −6.34334869438269873899055434856, −5.66648717686501125479789202683, −4.81320898258306963317840120039, −3.82905715967886507024272772360, −2.61703787488917174922692267020, −1.02744960940456294538773939700,
1.02744960940456294538773939700, 2.61703787488917174922692267020, 3.82905715967886507024272772360, 4.81320898258306963317840120039, 5.66648717686501125479789202683, 6.34334869438269873899055434856, 7.03236612956964786597528827910, 8.297910188101782088811239332122, 8.706928678145890375964307048640, 9.941876075251919766856521252555
