L(s) = 1 | + 0.445·2-s − 1.24·3-s − 0.801·4-s − 5-s − 0.554·6-s − 0.801·8-s + 0.554·9-s − 0.445·10-s + 11-s + 12-s + 1.80·13-s + 1.24·15-s + 0.445·16-s − 17-s + 0.246·18-s + 0.801·20-s + 0.445·22-s + 0.445·23-s + 24-s + 25-s + 0.801·26-s + 0.554·27-s − 0.445·29-s + 0.554·30-s + 32-s − 1.24·33-s − 0.445·34-s + ⋯ |
L(s) = 1 | + 0.445·2-s − 1.24·3-s − 0.801·4-s − 5-s − 0.554·6-s − 0.801·8-s + 0.554·9-s − 0.445·10-s + 11-s + 12-s + 1.80·13-s + 1.24·15-s + 0.445·16-s − 17-s + 0.246·18-s + 0.801·20-s + 0.445·22-s + 0.445·23-s + 24-s + 25-s + 0.801·26-s + 0.554·27-s − 0.445·29-s + 0.554·30-s + 32-s − 1.24·33-s − 0.445·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5759307871\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5759307871\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - 0.445T + T^{2} \) |
| 3 | \( 1 + 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.80T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.445T + T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.80T + T^{2} \) |
| 41 | \( 1 - 1.24T + T^{2} \) |
| 43 | \( 1 + 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.80T + T^{2} \) |
| 61 | \( 1 + 1.80T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 - 1.80T + T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 - 0.445T + T^{2} \) |
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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
−10.68361715949198551052520737933, −9.212042094562071598673421568414, −8.765093357142117301780734646110, −7.71618628165983957337333605697, −6.37108838349420513768591439135, −6.09771526591708339511341063994, −4.83159673742823251633254493699, −4.20261320699058776263071113448, −3.37208632723279301182684336793, −0.923170216322085595382892440582,
0.923170216322085595382892440582, 3.37208632723279301182684336793, 4.20261320699058776263071113448, 4.83159673742823251633254493699, 6.09771526591708339511341063994, 6.37108838349420513768591439135, 7.71618628165983957337333605697, 8.765093357142117301780734646110, 9.212042094562071598673421568414, 10.68361715949198551052520737933