| L(s) = 1 | − 546.·5-s − 1.68e4·7-s − 2.81e5·11-s − 1.34e6·13-s − 6.62e6·17-s + 2.91e6·19-s + 1.81e6·23-s − 4.85e7·25-s − 1.50e8·29-s + 7.40e7·31-s + 9.17e6·35-s + 6.30e8·37-s + 8.99e8·41-s − 1.77e9·43-s − 2.53e9·47-s + 2.82e8·49-s + 5.24e9·53-s + 1.53e8·55-s − 1.45e9·59-s − 9.10e9·61-s + 7.36e8·65-s + 2.71e9·67-s + 2.07e10·71-s − 3.66e9·73-s + 4.73e9·77-s − 3.89e10·79-s + 1.20e10·83-s + ⋯ |
| L(s) = 1 | − 0.0781·5-s − 0.377·7-s − 0.527·11-s − 1.00·13-s − 1.13·17-s + 0.269·19-s + 0.0587·23-s − 0.993·25-s − 1.36·29-s + 0.464·31-s + 0.0295·35-s + 1.49·37-s + 1.21·41-s − 1.84·43-s − 1.61·47-s + 0.142·49-s + 1.72·53-s + 0.0412·55-s − 0.265·59-s − 1.38·61-s + 0.0786·65-s + 0.245·67-s + 1.36·71-s − 0.206·73-s + 0.199·77-s − 1.42·79-s + 0.335·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.9119030019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9119030019\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 + 546.T + 4.88e7T^{2} \) |
| 11 | \( 1 + 2.81e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.34e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.62e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.91e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.81e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.50e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.40e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.30e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.99e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.77e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.53e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.24e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 1.45e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.10e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.71e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.07e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.66e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.89e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.80e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 3.23e10T + 7.15e21T^{2} \) |
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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
−9.970821705445431660732474985900, −9.316200069796563929336563002296, −8.091881969852787853810907442248, −7.25113590944263629299302964190, −6.20240181984775760417076289536, −5.10976573561237503916407295841, −4.09507309427502704882555246803, −2.84832949872905050259644975360, −1.92149128754055707236615878856, −0.38142822216168797357693117918,
0.38142822216168797357693117918, 1.92149128754055707236615878856, 2.84832949872905050259644975360, 4.09507309427502704882555246803, 5.10976573561237503916407295841, 6.20240181984775760417076289536, 7.25113590944263629299302964190, 8.091881969852787853810907442248, 9.316200069796563929336563002296, 9.970821705445431660732474985900
