| L(s) = 1 | − 7.03e3·5-s + 1.68e4·7-s + 1.56e4·11-s + 1.94e6·13-s + 1.06e7·17-s + 1.61e7·19-s − 1.30e7·23-s + 6.33e5·25-s − 1.29e8·29-s − 2.86e7·31-s − 1.18e8·35-s + 1.36e8·37-s − 5.64e7·41-s − 1.77e8·43-s + 3.18e8·47-s + 2.82e8·49-s − 4.84e9·53-s − 1.10e8·55-s − 3.10e9·59-s + 4.79e9·61-s − 1.36e10·65-s − 5.30e8·67-s + 8.63e9·71-s − 6.97e9·73-s + 2.63e8·77-s − 1.27e9·79-s + 5.73e10·83-s + ⋯ |
| L(s) = 1 | − 1.00·5-s + 0.377·7-s + 0.0293·11-s + 1.44·13-s + 1.81·17-s + 1.49·19-s − 0.423·23-s + 0.0129·25-s − 1.17·29-s − 0.179·31-s − 0.380·35-s + 0.323·37-s − 0.0761·41-s − 0.184·43-s + 0.202·47-s + 0.142·49-s − 1.59·53-s − 0.0295·55-s − 0.565·59-s + 0.727·61-s − 1.45·65-s − 0.0479·67-s + 0.568·71-s − 0.393·73-s + 0.0110·77-s − 0.0465·79-s + 1.59·83-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
| \(L(6)\) |
\(\approx\) |
\(2.297263568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.297263568\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 7.03e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 1.56e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.94e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.06e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.61e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.30e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.29e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.86e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.36e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.64e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.77e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 3.18e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.84e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.10e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 4.79e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 5.30e8T + 1.22e20T^{2} \) |
| 71 | \( 1 - 8.63e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 6.97e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.27e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.73e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.79e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 6.10e10T + 7.15e21T^{2} \) |
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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
−10.14036191355210732016704240537, −9.088764613012332813101479676520, −7.904850746704743708864454753347, −7.58980762907104791299400684691, −6.09129467212591627957771555193, −5.16359910692030962859220546214, −3.80572756772298525230976684461, −3.29527449183670678208090508900, −1.55503914575803081061177098736, −0.69013963680076872647308038887,
0.69013963680076872647308038887, 1.55503914575803081061177098736, 3.29527449183670678208090508900, 3.80572756772298525230976684461, 5.16359910692030962859220546214, 6.09129467212591627957771555193, 7.58980762907104791299400684691, 7.904850746704743708864454753347, 9.088764613012332813101479676520, 10.14036191355210732016704240537
