L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s + 2·11-s − 12-s − 2·13-s + 2·14-s − 15-s + 16-s − 18-s + 8·19-s + 20-s + 2·21-s − 2·22-s − 23-s + 24-s + 25-s + 2·26-s − 27-s − 2·28-s − 10·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.436·21-s − 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9058015745\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9058015745\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−10.19921733194512790300703139716, −9.660664297394421709821794550650, −9.075417416099267122163393331890, −7.71896779822359794181141522947, −7.00493605104502122471295939751, −6.08688989083657717123887285284, −5.31192173035210311351656132060, −3.83675961131181954206838591050, −2.51638204254903782188687395123, −0.942166573777626820274714022043,
0.942166573777626820274714022043, 2.51638204254903782188687395123, 3.83675961131181954206838591050, 5.31192173035210311351656132060, 6.08688989083657717123887285284, 7.00493605104502122471295939751, 7.71896779822359794181141522947, 9.075417416099267122163393331890, 9.660664297394421709821794550650, 10.19921733194512790300703139716