| L(s) = 1 | − 2-s + 4-s − 1.41·7-s − 8-s + 9-s + 1.41·13-s + 1.41·14-s + 16-s − 18-s + 1.41·23-s + 25-s − 1.41·26-s − 1.41·28-s − 1.41·29-s + 1.41·31-s − 32-s + 36-s − 1.41·37-s − 1.41·46-s + 1.00·49-s − 50-s + 1.41·52-s + 1.41·56-s + 1.41·58-s + 1.41·61-s − 1.41·62-s − 1.41·63-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 1.41·7-s − 8-s + 9-s + 1.41·13-s + 1.41·14-s + 16-s − 18-s + 1.41·23-s + 25-s − 1.41·26-s − 1.41·28-s − 1.41·29-s + 1.41·31-s − 32-s + 36-s − 1.41·37-s − 1.41·46-s + 1.00·49-s − 50-s + 1.41·52-s + 1.41·56-s + 1.41·58-s + 1.41·61-s − 1.41·62-s − 1.41·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6208365760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6208365760\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−10.51307679087487827394166142156, −9.684189188621603678925691110180, −9.063513557796564461620487439777, −8.219282158942985658340218874279, −6.86325262385508992743404148469, −6.75474156901699166104711484485, −5.52877907556120873266207610057, −3.83752665972184457084963138348, −2.91797017007895655644430298016, −1.25810018011888713064991706107,
1.25810018011888713064991706107, 2.91797017007895655644430298016, 3.83752665972184457084963138348, 5.52877907556120873266207610057, 6.75474156901699166104711484485, 6.86325262385508992743404148469, 8.219282158942985658340218874279, 9.063513557796564461620487439777, 9.684189188621603678925691110180, 10.51307679087487827394166142156
