| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 4·7-s + 8-s − 2·9-s + 11-s − 12-s − 13-s − 4·14-s + 16-s − 7·17-s − 2·18-s − 3·19-s + 4·21-s + 22-s − 24-s − 26-s + 5·27-s − 4·28-s − 4·29-s + 6·31-s + 32-s − 33-s − 7·34-s − 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 1.69·17-s − 0.471·18-s − 0.688·19-s + 0.872·21-s + 0.213·22-s − 0.204·24-s − 0.196·26-s + 0.962·27-s − 0.755·28-s − 0.742·29-s + 1.07·31-s + 0.176·32-s − 0.174·33-s − 1.20·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.33981883717841530698728832250, −9.306420860153218895558185083831, −8.470593426325214029406055436314, −6.90149710443616766187323727244, −6.50957455206104884247861007412, −5.66074294846131677349398169974, −4.55255106907027843064175239055, −3.46602077604760174991666093933, −2.38389183928965643037856764307, 0,
2.38389183928965643037856764307, 3.46602077604760174991666093933, 4.55255106907027843064175239055, 5.66074294846131677349398169974, 6.50957455206104884247861007412, 6.90149710443616766187323727244, 8.470593426325214029406055436314, 9.306420860153218895558185083831, 10.33981883717841530698728832250
