| L(s) = 1 | + 31.6·2-s + 243·3-s − 1.04e3·4-s − 7.92e3·5-s + 7.68e3·6-s + 5.53e4·7-s − 9.79e4·8-s + 5.90e4·9-s − 2.50e5·10-s + 7.49e5·11-s − 2.54e5·12-s − 8.55e5·13-s + 1.75e6·14-s − 1.92e6·15-s − 9.49e5·16-s − 8.42e6·17-s + 1.86e6·18-s − 2.35e5·19-s + 8.30e6·20-s + 1.34e7·21-s + 2.36e7·22-s + 1.10e7·23-s − 2.37e7·24-s + 1.39e7·25-s − 2.70e7·26-s + 1.43e7·27-s − 5.80e7·28-s + ⋯ |
| L(s) = 1 | + 0.698·2-s + 0.577·3-s − 0.511·4-s − 1.13·5-s + 0.403·6-s + 1.24·7-s − 1.05·8-s + 0.333·9-s − 0.791·10-s + 1.40·11-s − 0.295·12-s − 0.638·13-s + 0.870·14-s − 0.654·15-s − 0.226·16-s − 1.43·17-s + 0.232·18-s − 0.0218·19-s + 0.580·20-s + 0.719·21-s + 0.980·22-s + 0.357·23-s − 0.609·24-s + 0.284·25-s − 0.446·26-s + 0.192·27-s − 0.637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.769759382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.769759382\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 31.6T + 2.04e3T^{2} \) |
| 5 | \( 1 + 7.92e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 5.53e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 7.49e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 8.55e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 8.42e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 2.35e5T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.10e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 3.07e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 9.66e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.15e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.96e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.69e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 3.75e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.44e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.16e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 3.45e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.89e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.54e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.21e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.97e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.65e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.01e11T + 7.15e21T^{2} \) |
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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
−10.99398491375554399671910160337, −9.325208000358727540380491190930, −8.651321641995834621344550203332, −7.74582716014402043975790993189, −6.60352347264077638542309414113, −4.95164823890063357582261278513, −4.28850106533574590481381248733, −3.55466895895539271588440344550, −2.08097997539639206064184629703, −0.66445915179312607369970364398,
0.66445915179312607369970364398, 2.08097997539639206064184629703, 3.55466895895539271588440344550, 4.28850106533574590481381248733, 4.95164823890063357582261278513, 6.60352347264077638542309414113, 7.74582716014402043975790993189, 8.651321641995834621344550203332, 9.325208000358727540380491190930, 10.99398491375554399671910160337
