| L(s) = 1 | − 12.2·3-s + 625·5-s + 9.62e3·7-s − 1.95e4·9-s − 5.56e4·11-s − 1.69e5·13-s − 7.63e3·15-s + 2.07e5·17-s − 8.02e5·19-s − 1.17e5·21-s − 1.24e6·23-s + 3.90e5·25-s + 4.78e5·27-s + 4.28e6·29-s − 3.58e6·31-s + 6.79e5·33-s + 6.01e6·35-s + 2.89e6·37-s + 2.07e6·39-s + 2.51e7·41-s + 2.00e7·43-s − 1.22e7·45-s + 3.73e7·47-s + 5.22e7·49-s − 2.53e6·51-s + 2.55e7·53-s − 3.47e7·55-s + ⋯ |
| L(s) = 1 | − 0.0870·3-s + 0.447·5-s + 1.51·7-s − 0.992·9-s − 1.14·11-s − 1.64·13-s − 0.0389·15-s + 0.602·17-s − 1.41·19-s − 0.131·21-s − 0.925·23-s + 0.200·25-s + 0.173·27-s + 1.12·29-s − 0.697·31-s + 0.0997·33-s + 0.677·35-s + 0.254·37-s + 0.143·39-s + 1.39·41-s + 0.893·43-s − 0.443·45-s + 1.11·47-s + 1.29·49-s − 0.0524·51-s + 0.444·53-s − 0.512·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.780657589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780657589\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| good | 3 | \( 1 + 12.2T + 1.96e4T^{2} \) |
| 7 | \( 1 - 9.62e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.56e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.69e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.07e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.02e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.58e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.89e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.51e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.73e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.55e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.96e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.09e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.31e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.81e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.39e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.23e8T + 7.60e17T^{2} \) |
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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
−10.31068081492856160669760320995, −9.052420579470327006039875130690, −8.075695727370627389736340057313, −7.51941877510880652049208696111, −5.97936205691988454339381908690, −5.19761889009890666409884269644, −4.42250499900979270985689532273, −2.60352041334856547502443095131, −2.09550557360685908673031799945, −0.55636167118956314232227974377,
0.55636167118956314232227974377, 2.09550557360685908673031799945, 2.60352041334856547502443095131, 4.42250499900979270985689532273, 5.19761889009890666409884269644, 5.97936205691988454339381908690, 7.51941877510880652049208696111, 8.075695727370627389736340057313, 9.052420579470327006039875130690, 10.31068081492856160669760320995
