L(s) = 1 | + 3-s + 4·5-s − 2·7-s + 9-s − 11-s + 4·15-s − 6·17-s + 4·19-s − 2·21-s − 6·23-s + 11·25-s + 27-s + 6·29-s − 33-s − 8·35-s + 6·37-s − 10·41-s − 8·43-s + 4·45-s + 6·47-s − 3·49-s − 6·51-s − 12·53-s − 4·55-s + 4·57-s − 8·59-s + 4·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.03·15-s − 1.45·17-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s − 0.174·33-s − 1.35·35-s + 0.986·37-s − 1.56·41-s − 1.21·43-s + 0.596·45-s + 0.875·47-s − 3/7·49-s − 0.840·51-s − 1.64·53-s − 0.539·55-s + 0.529·57-s − 1.04·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.765930747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.765930747\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.16989805773731180913070741133, −10.69403003343854619075048453503, −9.829548973232970908437928174650, −9.356494453524738574521049972109, −8.269419736966949478022139590048, −6.76400139406901473566951722668, −6.07683367095826721520266428891, −4.80679270732192266832740349851, −3.05003123994539200255449512211, −1.94084613584153475610096907172,
1.94084613584153475610096907172, 3.05003123994539200255449512211, 4.80679270732192266832740349851, 6.07683367095826721520266428891, 6.76400139406901473566951722668, 8.269419736966949478022139590048, 9.356494453524738574521049972109, 9.829548973232970908437928174650, 10.69403003343854619075048453503, 12.16989805773731180913070741133