L(s) = 1 | − 3-s + 2·5-s + 9-s + 11-s + 2·13-s − 2·15-s + 6·17-s + 4·23-s − 25-s − 27-s + 2·29-s − 33-s − 10·37-s − 2·39-s + 6·41-s − 8·43-s + 2·45-s − 4·47-s − 7·49-s − 6·51-s − 6·53-s + 2·55-s − 12·59-s + 2·61-s + 4·65-s + 4·67-s − 4·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.174·33-s − 1.64·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s + 0.298·45-s − 0.583·47-s − 49-s − 0.840·51-s − 0.824·53-s + 0.269·55-s − 1.56·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s − 0.481·69-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
\(L(1)\) |
\(\approx\) |
\(1.276994191\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276994191\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−11.97559276395264382268349706705, −10.97073736251352258496300508621, −10.08030090780381353518984735580, −9.305007869584320081543641621705, −8.065545776521386769835087972572, −6.78625245992293517864906809126, −5.88581426070419577004157985232, −4.97830796048395948436336588916, −3.36387077635425811672598608724, −1.49507921150573825532250127010,
1.49507921150573825532250127010, 3.36387077635425811672598608724, 4.97830796048395948436336588916, 5.88581426070419577004157985232, 6.78625245992293517864906809126, 8.065545776521386769835087972572, 9.305007869584320081543641621705, 10.08030090780381353518984735580, 10.97073736251352258496300508621, 11.97559276395264382268349706705