L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 3·7-s + 8-s + 9-s − 11-s + 12-s − 3·13-s − 3·14-s + 16-s + 18-s − 5·19-s − 3·21-s − 22-s + 23-s + 24-s − 3·26-s + 27-s − 3·28-s − 9·29-s − 2·31-s + 32-s − 33-s + 36-s − 5·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.832·13-s − 0.801·14-s + 1/4·16-s + 0.235·18-s − 1.14·19-s − 0.654·21-s − 0.213·22-s + 0.208·23-s + 0.204·24-s − 0.588·26-s + 0.192·27-s − 0.566·28-s − 1.67·29-s − 0.359·31-s + 0.176·32-s − 0.174·33-s + 1/6·36-s − 0.811·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.064343597538035825315431255449, −7.39533477839573005025027824615, −6.69456593733697678146438723218, −5.99675305780424268417178538589, −5.11889207917423108405808321550, −4.22442363005742196881444759444, −3.46561454007586113991876104892, −2.71198064437539097287092651177, −1.87134268487925037776431244015, 0,
1.87134268487925037776431244015, 2.71198064437539097287092651177, 3.46561454007586113991876104892, 4.22442363005742196881444759444, 5.11889207917423108405808321550, 5.99675305780424268417178538589, 6.69456593733697678146438723218, 7.39533477839573005025027824615, 8.064343597538035825315431255449