| L(s) = 1 | + 79.6·3-s + 1.42e3·5-s − 2.40e3·7-s − 1.33e4·9-s − 6.93e4·11-s + 1.05e5·13-s + 1.13e5·15-s + 5.68e5·17-s + 3.96e5·19-s − 1.91e5·21-s + 6.20e5·23-s + 7.37e4·25-s − 2.63e6·27-s + 4.87e6·29-s + 1.42e6·31-s − 5.52e6·33-s − 3.41e6·35-s + 1.31e7·37-s + 8.43e6·39-s − 2.03e7·41-s + 1.11e7·43-s − 1.89e7·45-s + 1.99e7·47-s + 5.76e6·49-s + 4.52e7·51-s + 5.65e7·53-s − 9.87e7·55-s + ⋯ |
| L(s) = 1 | + 0.567·3-s + 1.01·5-s − 0.377·7-s − 0.677·9-s − 1.42·11-s + 1.02·13-s + 0.578·15-s + 1.65·17-s + 0.697·19-s − 0.214·21-s + 0.462·23-s + 0.0377·25-s − 0.952·27-s + 1.28·29-s + 0.277·31-s − 0.810·33-s − 0.385·35-s + 1.14·37-s + 0.584·39-s − 1.12·41-s + 0.498·43-s − 0.690·45-s + 0.595·47-s + 0.142·49-s + 0.936·51-s + 0.983·53-s − 1.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.956036891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.956036891\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 - 79.6T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.42e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 6.93e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.05e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.96e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.20e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.87e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.42e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.31e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.03e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.99e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.65e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.09e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.20e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.02e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.70e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.82e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.47e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.09e9T + 7.60e17T^{2} \) |
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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
−11.90557305644239209878189761265, −10.51577814546798453348369595692, −9.774044767825023060481267151363, −8.629485015532953666740537053133, −7.67247765194062187349982794685, −6.05355998096929881896557211237, −5.31084017736087092700882279305, −3.32759639476782011177223208765, −2.47917126956516586453061634982, −0.931443101815477987428082334087,
0.931443101815477987428082334087, 2.47917126956516586453061634982, 3.32759639476782011177223208765, 5.31084017736087092700882279305, 6.05355998096929881896557211237, 7.67247765194062187349982794685, 8.629485015532953666740537053133, 9.774044767825023060481267151363, 10.51577814546798453348369595692, 11.90557305644239209878189761265
