L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s − 1.23·7-s − 2.23·8-s + 9-s − 2·11-s − 0.618·12-s + 13-s − 2.00·14-s − 4.85·16-s + 7.47·17-s + 1.61·18-s − 2.23·19-s + 1.23·21-s − 3.23·22-s + 4·23-s + 2.23·24-s + 1.61·26-s − 27-s − 0.763·28-s + 1.23·29-s − 2·31-s − 3.38·32-s + 2·33-s + 12.0·34-s + ⋯ |
L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s − 0.467·7-s − 0.790·8-s + 0.333·9-s − 0.603·11-s − 0.178·12-s + 0.277·13-s − 0.534·14-s − 1.21·16-s + 1.81·17-s + 0.381·18-s − 0.512·19-s + 0.269·21-s − 0.689·22-s + 0.834·23-s + 0.456·24-s + 0.317·26-s − 0.192·27-s − 0.144·28-s + 0.229·29-s − 0.359·31-s − 0.597·32-s + 0.348·33-s + 2.07·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 7.94T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 - 3.70T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.23364478070541773066492035082, −6.53358138018195434591214884713, −5.93019035014758498409729296383, −5.28981632039806019278882603763, −4.86665660378284655692390599575, −3.90534058451608448728005536771, −3.31157694363779819151537939095, −2.59807439228722807123824730793, −1.23800182648834983810266662841, 0,
1.23800182648834983810266662841, 2.59807439228722807123824730793, 3.31157694363779819151537939095, 3.90534058451608448728005536771, 4.86665660378284655692390599575, 5.28981632039806019278882603763, 5.93019035014758498409729296383, 6.53358138018195434591214884713, 7.23364478070541773066492035082