L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 2·11-s − 12-s + 14-s + 16-s − 18-s + 19-s + 21-s − 2·22-s + 24-s − 27-s − 28-s − 29-s + 8·31-s − 32-s − 2·33-s + 36-s + 2·37-s − 38-s − 2·41-s − 42-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 + 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.229·19-s + 0.218·21-s − 0.426·22-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.185·29-s + 1.43·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s + 0.328·37-s − 0.162·38-s − 0.312·41-s − 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−12.88906606217271, −12.22855445832998, −11.89401181932252, −11.55158906562540, −11.11150259361870, −10.53747122640910, −10.06142268643654, −9.835123389883542, −9.256695651003016, −8.837936203748012, −8.257336701390581, −7.905237995114159, −7.233663536446359, −6.826956936180309, −6.413916868000123, −6.020762121097214, −5.412607783451434, −4.872988560958841, −4.354088154264665, −3.626157733512810, −3.298786464544512, −2.457128403470922, −2.024086324360006, −1.164229383677689, −0.8028774596763561, 0,
0.8028774596763561, 1.164229383677689, 2.024086324360006, 2.457128403470922, 3.298786464544512, 3.626157733512810, 4.354088154264665, 4.872988560958841, 5.412607783451434, 6.020762121097214, 6.413916868000123, 6.826956936180309, 7.233663536446359, 7.905237995114159, 8.257336701390581, 8.837936203748012, 9.256695651003016, 9.835123389883542, 10.06142268643654, 10.53747122640910, 11.11150259361870, 11.55158906562540, 11.89401181932252, 12.22855445832998, 12.88906606217271