L(s) = 1 | − 3-s − 5-s + 2·7-s − 2·9-s + 2·11-s + 6·13-s + 15-s − 7·17-s − 5·19-s − 2·21-s − 4·23-s + 25-s + 5·27-s − 31-s − 2·33-s − 2·35-s + 7·37-s − 6·39-s − 3·41-s + 9·43-s + 2·45-s + 2·47-s − 3·49-s + 7·51-s − 9·53-s − 2·55-s + 5·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s − 2/3·9-s + 0.603·11-s + 1.66·13-s + 0.258·15-s − 1.69·17-s − 1.14·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.962·27-s − 0.179·31-s − 0.348·33-s − 0.338·35-s + 1.15·37-s − 0.960·39-s − 0.468·41-s + 1.37·43-s + 0.298·45-s + 0.291·47-s − 3/7·49-s + 0.980·51-s − 1.23·53-s − 0.269·55-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.300831000\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.300831000\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.80748023948079028052069422650, −6.72118979373283634357281266680, −6.29382816024444541571417613592, −5.80771105926536425072144034460, −4.78043583400838437681208058286, −4.24582911113100366632537271782, −3.64279525703506430240713372675, −2.49105804464989092413224557078, −1.65023346940688635405322341475, −0.56354025057412059849396502926,
0.56354025057412059849396502926, 1.65023346940688635405322341475, 2.49105804464989092413224557078, 3.64279525703506430240713372675, 4.24582911113100366632537271782, 4.78043583400838437681208058286, 5.80771105926536425072144034460, 6.29382816024444541571417613592, 6.72118979373283634357281266680, 7.80748023948079028052069422650