L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 5·7-s − 2·10-s + 2·11-s − 6·13-s − 10·14-s − 4·16-s − 17-s + 2·19-s − 2·20-s + 4·22-s + 23-s + 25-s − 12·26-s − 10·28-s + 29-s − 5·31-s − 8·32-s − 2·34-s + 5·35-s − 7·37-s + 4·38-s + 7·41-s − 8·43-s + 4·44-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 1.88·7-s − 0.632·10-s + 0.603·11-s − 1.66·13-s − 2.67·14-s − 16-s − 0.242·17-s + 0.458·19-s − 0.447·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s − 2.35·26-s − 1.88·28-s + 0.185·29-s − 0.898·31-s − 1.41·32-s − 0.342·34-s + 0.845·35-s − 1.15·37-s + 0.648·38-s + 1.09·41-s − 1.21·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−9.520030452769672320712627107087, −8.934048176077265695872190702403, −7.31102952580309465306532731292, −6.86142089136040417683964957289, −5.98288576571748361818329177001, −5.09424776006149075225955854265, −4.08727238312554578444301288677, −3.33418197528540395822473692124, −2.54234164036259966688798016490, 0,
2.54234164036259966688798016490, 3.33418197528540395822473692124, 4.08727238312554578444301288677, 5.09424776006149075225955854265, 5.98288576571748361818329177001, 6.86142089136040417683964957289, 7.31102952580309465306532731292, 8.934048176077265695872190702403, 9.520030452769672320712627107087