| L(s) = 1 | + 32·2-s − 243·3-s + 1.02e3·4-s + 1.23e4·5-s − 7.77e3·6-s + 1.68e4·7-s + 3.27e4·8-s + 5.90e4·9-s + 3.93e5·10-s − 3.52e5·11-s − 2.48e5·12-s + 1.58e6·13-s + 5.37e5·14-s − 2.98e6·15-s + 1.04e6·16-s − 2.48e6·17-s + 1.88e6·18-s − 1.61e7·19-s + 1.25e7·20-s − 4.08e6·21-s − 1.12e7·22-s + 4.52e7·23-s − 7.96e6·24-s + 1.02e8·25-s + 5.05e7·26-s − 1.43e7·27-s + 1.72e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.76·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.24·10-s − 0.660·11-s − 0.288·12-s + 1.18·13-s + 0.267·14-s − 1.01·15-s + 0.250·16-s − 0.425·17-s + 0.235·18-s − 1.49·19-s + 0.880·20-s − 0.218·21-s − 0.466·22-s + 1.46·23-s − 0.204·24-s + 2.09·25-s + 0.834·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.848655353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.848655353\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 3 | \( 1 + 243T \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 5 | \( 1 - 1.23e4T + 4.88e7T^{2} \) |
| 11 | \( 1 + 3.52e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.58e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.48e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.61e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.52e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.70e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.42e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.89e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 2.36e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.90e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.17e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 6.05e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.82e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.02e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.64e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 6.23e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.55e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.48e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.22e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.88e10T + 7.15e21T^{2} \) |
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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
−13.39182727332886704911545079992, −12.81998979725578971442621519433, −11.06203687288084931414208563302, −10.33100625695939919295291250616, −8.735837472892578776949479334125, −6.64088832035783316296017083533, −5.79837116350532784342635439682, −4.67360540396806892900630579014, −2.55330289628493633068920698053, −1.27729046993793957084610737905,
1.27729046993793957084610737905, 2.55330289628493633068920698053, 4.67360540396806892900630579014, 5.79837116350532784342635439682, 6.64088832035783316296017083533, 8.735837472892578776949479334125, 10.33100625695939919295291250616, 11.06203687288084931414208563302, 12.81998979725578971442621519433, 13.39182727332886704911545079992
