| L(s) = 1 | − 243·3-s + 9.00e3·5-s − 3.34e4·7-s + 5.90e4·9-s + 1.14e4·11-s + 4.24e5·13-s − 2.18e6·15-s + 7.76e6·17-s − 7.16e6·19-s + 8.13e6·21-s + 1.00e7·23-s + 3.21e7·25-s − 1.43e7·27-s + 3.12e7·29-s + 5.25e6·31-s − 2.77e6·33-s − 3.01e8·35-s − 5.58e8·37-s − 1.03e8·39-s − 2.54e8·41-s − 1.36e9·43-s + 5.31e8·45-s + 2.16e9·47-s − 8.55e8·49-s − 1.88e9·51-s + 3.93e9·53-s + 1.02e8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.28·5-s − 0.753·7-s + 0.333·9-s + 0.0213·11-s + 0.317·13-s − 0.743·15-s + 1.32·17-s − 0.663·19-s + 0.434·21-s + 0.326·23-s + 0.659·25-s − 0.192·27-s + 0.283·29-s + 0.0329·31-s − 0.0123·33-s − 0.970·35-s − 1.32·37-s − 0.183·39-s − 0.342·41-s − 1.41·43-s + 0.429·45-s + 1.37·47-s − 0.432·49-s − 0.765·51-s + 1.29·53-s + 0.0275·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.101322531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101322531\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 243T \) |
| good | 5 | \( 1 - 9.00e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 3.34e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.14e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 4.24e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 7.76e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 7.16e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.00e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 3.12e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 5.25e6T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.58e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.54e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.36e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.16e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.93e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 9.48e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.90e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 9.83e8T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.06e8T + 2.31e20T^{2} \) |
| 73 | \( 1 + 5.83e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.66e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.82e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.93e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.09e11T + 7.15e21T^{2} \) |
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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
−11.84014743153840496110605154982, −10.38687164958418241921652309935, −9.897054741104057305312979425354, −8.674200560265444584280320338854, −6.98320377858196299766975183603, −6.04359883428603019319059323183, −5.21015575305968872995988220142, −3.49966320875282294057599452724, −2.04274115546493721689419606595, −0.78078682021587446595477366146,
0.78078682021587446595477366146, 2.04274115546493721689419606595, 3.49966320875282294057599452724, 5.21015575305968872995988220142, 6.04359883428603019319059323183, 6.98320377858196299766975183603, 8.674200560265444584280320338854, 9.897054741104057305312979425354, 10.38687164958418241921652309935, 11.84014743153840496110605154982
