L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 2·9-s − 5·11-s − 12-s − 5·13-s + 14-s + 16-s − 2·18-s − 6·19-s − 21-s − 5·22-s + 23-s − 24-s − 5·25-s − 5·26-s + 5·27-s + 28-s + 6·29-s + 8·31-s + 32-s + 5·33-s − 2·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 1.50·11-s − 0.288·12-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.471·18-s − 1.37·19-s − 0.218·21-s − 1.06·22-s + 0.208·23-s − 0.204·24-s − 25-s − 0.980·26-s + 0.962·27-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.870·33-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.919580205497809214721828176561, −8.426981763238328825777480614947, −7.902476972027407870307469853024, −6.79642318501152767639984710260, −5.98297584186990174254768513700, −5.00558535884058916147640308815, −4.65621481774096026687557687085, −3.02811664827220844560879586853, −2.20099778339682437421938450648, 0,
2.20099778339682437421938450648, 3.02811664827220844560879586853, 4.65621481774096026687557687085, 5.00558535884058916147640308815, 5.98297584186990174254768513700, 6.79642318501152767639984710260, 7.902476972027407870307469853024, 8.426981763238328825777480614947, 9.919580205497809214721828176561