L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s − 2·9-s − 11-s + 12-s + 5·13-s + 16-s + 6·17-s − 2·18-s + 2·19-s − 22-s + 6·23-s + 24-s − 5·25-s + 5·26-s − 5·27-s + 3·29-s + 8·31-s + 32-s − 33-s + 6·34-s − 2·36-s + 2·37-s + 2·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.301·11-s + 0.288·12-s + 1.38·13-s + 1/4·16-s + 1.45·17-s − 0.471·18-s + 0.458·19-s − 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s + 0.980·26-s − 0.962·27-s + 0.557·29-s + 1.43·31-s + 0.176·32-s − 0.174·33-s + 1.02·34-s − 1/3·36-s + 0.328·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.118118033\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.118118033\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.937154387927371662517251041033, −8.933030074235662820580600918861, −8.164909575981619811322020495480, −7.49306261678924822797408508942, −6.25502001754230582135037303450, −5.66678666347906696017766800720, −4.62115575450762314305718823399, −3.37668246136945221572009714265, −2.94788694044649373429866243305, −1.37229172918044571298512618026,
1.37229172918044571298512618026, 2.94788694044649373429866243305, 3.37668246136945221572009714265, 4.62115575450762314305718823399, 5.66678666347906696017766800720, 6.25502001754230582135037303450, 7.49306261678924822797408508942, 8.164909575981619811322020495480, 8.933030074235662820580600918861, 9.937154387927371662517251041033