| L(s) = 1 | + 4-s − 0.445·5-s + 9-s + 1.24·13-s + 16-s − 0.445·20-s − 1.80·23-s − 0.801·25-s − 1.80·31-s + 36-s + 1.24·41-s + 1.24·43-s − 0.445·45-s + 49-s + 1.24·52-s + 64-s − 0.554·65-s + 1.24·71-s − 0.445·80-s + 81-s − 1.80·83-s − 1.80·89-s − 1.80·92-s − 0.801·100-s − 0.445·101-s − 0.445·103-s − 0.445·109-s + ⋯ |
| L(s) = 1 | + 4-s − 0.445·5-s + 9-s + 1.24·13-s + 16-s − 0.445·20-s − 1.80·23-s − 0.801·25-s − 1.80·31-s + 36-s + 1.24·41-s + 1.24·43-s − 0.445·45-s + 49-s + 1.24·52-s + 64-s − 0.554·65-s + 1.24·71-s − 0.445·80-s + 81-s − 1.80·83-s − 1.80·89-s − 1.80·92-s − 0.801·100-s − 0.445·101-s − 0.445·103-s − 0.445·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.504006919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504006919\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2011 | \( 1+O(T) \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.24T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.80T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.24T + T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.24T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.80T + T^{2} \) |
| 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
−9.433165867166389605100889166681, −8.383017311307391300186212949089, −7.64499294080361593415047298585, −7.13914371461768728472912077340, −6.14374611519011938166547855601, −5.63800325129470412524547640051, −4.07556281147085956434886199465, −3.75370341881959017979953493370, −2.34543493833801924716716633503, −1.40324453744509293915956313039,
1.40324453744509293915956313039, 2.34543493833801924716716633503, 3.75370341881959017979953493370, 4.07556281147085956434886199465, 5.63800325129470412524547640051, 6.14374611519011938166547855601, 7.13914371461768728472912077340, 7.64499294080361593415047298585, 8.383017311307391300186212949089, 9.433165867166389605100889166681
