L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 4·7-s + 8-s + 9-s − 10-s + 12-s + 13-s − 4·14-s − 15-s + 16-s + 2·17-s + 18-s − 4·19-s − 20-s − 4·21-s + 8·23-s + 24-s + 25-s + 26-s + 27-s − 4·28-s − 2·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s − 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.85987072730676, −14.52930546074750, −13.59676701652196, −13.18175786457411, −13.02986694632963, −12.41127551655673, −11.99578774037004, −11.14462611291151, −10.84564499988848, −10.15212092620135, −9.633083163702873, −9.064378407255781, −8.604420354838946, −7.934559284457693, −7.105837974589088, −6.987217664062354, −6.319913803919769, −5.643149477055904, −5.116480594771980, −4.198919997399951, −3.815482466125074, −3.191564440407635, −2.823875090125015, −1.994569303907133, −1.028349566406169, 0,
1.028349566406169, 1.994569303907133, 2.823875090125015, 3.191564440407635, 3.815482466125074, 4.198919997399951, 5.116480594771980, 5.643149477055904, 6.319913803919769, 6.987217664062354, 7.105837974589088, 7.934559284457693, 8.604420354838946, 9.064378407255781, 9.633083163702873, 10.15212092620135, 10.84564499988848, 11.14462611291151, 11.99578774037004, 12.41127551655673, 13.02986694632963, 13.18175786457411, 13.59676701652196, 14.52930546074750, 14.85987072730676