L(s) = 1 | + 3-s + 5-s + 7-s − 2·9-s + 3·11-s − 13-s + 15-s − 3·17-s + 2·19-s + 21-s − 6·23-s + 25-s − 5·27-s − 9·29-s + 8·31-s + 3·33-s + 35-s − 10·37-s − 39-s + 2·43-s − 2·45-s − 3·47-s + 49-s − 3·51-s + 3·55-s + 2·57-s + 12·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 1.43·31-s + 0.522·33-s + 0.169·35-s − 1.64·37-s − 0.160·39-s + 0.304·43-s − 0.298·45-s − 0.437·47-s + 1/7·49-s − 0.420·51-s + 0.404·55-s + 0.264·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.344301542\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.344301542\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.43051342969939403565134696798, −12.06507020707332784174112854689, −11.23967367108056747780641464517, −9.891670406720994521533105632335, −8.980861688416760643046869733716, −8.046356379591346049691472796650, −6.67050874393051432711601966688, −5.40018100386713622724414679311, −3.80030272906745320072379387841, −2.14587146010097196258543648767,
2.14587146010097196258543648767, 3.80030272906745320072379387841, 5.40018100386713622724414679311, 6.67050874393051432711601966688, 8.046356379591346049691472796650, 8.980861688416760643046869733716, 9.891670406720994521533105632335, 11.23967367108056747780641464517, 12.06507020707332784174112854689, 13.43051342969939403565134696798