| L(s) = 1 | − 1.24·2-s + 0.554·4-s + 0.445·5-s + 0.554·8-s − 0.554·10-s − 1.24·16-s + 1.24·19-s + 0.246·20-s − 0.801·25-s + 1.80·29-s + 0.999·32-s + 1.24·37-s − 1.55·38-s + 0.246·40-s − 0.445·43-s + 49-s + 50-s − 2.24·58-s − 71-s − 0.445·73-s − 1.55·74-s + 0.692·76-s − 0.445·79-s − 0.554·80-s − 1.24·83-s + 0.554·86-s + 1.80·89-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.554·4-s + 0.445·5-s + 0.554·8-s − 0.554·10-s − 1.24·16-s + 1.24·19-s + 0.246·20-s − 0.801·25-s + 1.80·29-s + 0.999·32-s + 1.24·37-s − 1.55·38-s + 0.246·40-s − 0.445·43-s + 49-s + 50-s − 2.24·58-s − 71-s − 0.445·73-s − 1.55·74-s + 0.692·76-s − 0.445·79-s − 0.554·80-s − 1.24·83-s + 0.554·86-s + 1.80·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5406311020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5406311020\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 1.24T + T^{2} \) |
| 5 | \( 1 - 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.80T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.24T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 + 1.24T + T^{2} \) |
| 89 | \( 1 - 1.80T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43058593266493346706313290487, −9.833420775873518817936336720201, −9.170241421950823689049236844172, −8.264653899015990838034506744415, −7.53479988776860593174990930549, −6.57025284206264928971284169702, −5.43620883403123476947121773662, −4.28360347967664439419897041164, −2.69641103570264554325485886409, −1.26584969481661945262580678838,
1.26584969481661945262580678838, 2.69641103570264554325485886409, 4.28360347967664439419897041164, 5.43620883403123476947121773662, 6.57025284206264928971284169702, 7.53479988776860593174990930549, 8.264653899015990838034506744415, 9.170241421950823689049236844172, 9.833420775873518817936336720201, 10.43058593266493346706313290487
