| L(s) = 1 | + 1.80·2-s + 2.24·4-s − 7-s + 2.24·8-s + 1.80·13-s − 1.80·14-s + 1.80·16-s − 0.445·17-s + 0.445·19-s − 1.24·23-s + 25-s + 3.24·26-s − 2.24·28-s + 1.00·32-s − 0.801·34-s − 1.80·37-s + 0.801·38-s + 41-s − 0.445·43-s − 2.24·46-s − 1.80·47-s + 49-s + 1.80·50-s + 4.04·52-s − 2.24·56-s − 68-s − 3.24·74-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.24·4-s − 7-s + 2.24·8-s + 1.80·13-s − 1.80·14-s + 1.80·16-s − 0.445·17-s + 0.445·19-s − 1.24·23-s + 25-s + 3.24·26-s − 2.24·28-s + 1.00·32-s − 0.801·34-s − 1.80·37-s + 0.801·38-s + 41-s − 0.445·43-s − 2.24·46-s − 1.80·47-s + 49-s + 1.80·50-s + 4.04·52-s − 2.24·56-s − 68-s − 3.24·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.351552127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.351552127\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.80T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.80T + T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 - 0.445T + T^{2} \) |
| 23 | \( 1 + 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 + 1.80T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.80T + T^{2} \) |
| 97 | \( 1 + 1.24T + T^{2} \) |
| 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
−9.011038301034964812809647599280, −8.194639787662539846992120698394, −7.04800402396822059485308327285, −6.48224018561616553590788887848, −5.94455202079573899479751621260, −5.17627977169377149822466210185, −4.14128133527663133054093708643, −3.55129377263065664685285013559, −2.88269371753958251897181709838, −1.64430122150954536375915455795,
1.64430122150954536375915455795, 2.88269371753958251897181709838, 3.55129377263065664685285013559, 4.14128133527663133054093708643, 5.17627977169377149822466210185, 5.94455202079573899479751621260, 6.48224018561616553590788887848, 7.04800402396822059485308327285, 8.194639787662539846992120698394, 9.011038301034964812809647599280
