L(s) = 1 | − 3·3-s + 2·5-s + 3·7-s + 6·9-s + 2·11-s − 3·13-s − 6·15-s − 17-s − 9·21-s − 5·23-s − 25-s − 9·27-s − 3·29-s + 6·31-s − 6·33-s + 6·35-s + 6·37-s + 9·39-s + 12·41-s + 10·43-s + 12·45-s + 8·47-s + 2·49-s + 3·51-s − 3·53-s + 4·55-s − 3·59-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.894·5-s + 1.13·7-s + 2·9-s + 0.603·11-s − 0.832·13-s − 1.54·15-s − 0.242·17-s − 1.96·21-s − 1.04·23-s − 1/5·25-s − 1.73·27-s − 0.557·29-s + 1.07·31-s − 1.04·33-s + 1.01·35-s + 0.986·37-s + 1.44·39-s + 1.87·41-s + 1.52·43-s + 1.78·45-s + 1.16·47-s + 2/7·49-s + 0.420·51-s − 0.412·53-s + 0.539·55-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.452358602\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.452358602\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.75368118288849823576512395149, −7.43780823074750434387955411160, −6.28968590721013748123072752107, −6.02040759171744991110992200526, −5.38490275424350212682952082403, −4.51730300197003160097237112241, −4.23401860961852148122002527301, −2.46330417017491156419222564367, −1.65522353260613738853776601855, −0.73310587238049562800200747900,
0.73310587238049562800200747900, 1.65522353260613738853776601855, 2.46330417017491156419222564367, 4.23401860961852148122002527301, 4.51730300197003160097237112241, 5.38490275424350212682952082403, 6.02040759171744991110992200526, 6.28968590721013748123072752107, 7.43780823074750434387955411160, 7.75368118288849823576512395149