L(s) = 1 | − 10.6·2-s − 243·3-s − 1.93e3·4-s − 9.08e3·5-s + 2.58e3·6-s + 5.76e4·7-s + 4.23e4·8-s + 5.90e4·9-s + 9.65e4·10-s + 8.14e5·11-s + 4.70e5·12-s + 2.09e6·13-s − 6.12e5·14-s + 2.20e6·15-s + 3.51e6·16-s − 4.28e5·17-s − 6.27e5·18-s + 2.07e7·19-s + 1.75e7·20-s − 1.40e7·21-s − 8.65e6·22-s − 2.39e7·23-s − 1.02e7·24-s + 3.36e7·25-s − 2.23e7·26-s − 1.43e7·27-s − 1.11e8·28-s + ⋯ |
L(s) = 1 | − 0.234·2-s − 0.577·3-s − 0.944·4-s − 1.29·5-s + 0.135·6-s + 1.29·7-s + 0.456·8-s + 0.333·9-s + 0.305·10-s + 1.52·11-s + 0.545·12-s + 1.56·13-s − 0.304·14-s + 0.750·15-s + 0.837·16-s − 0.0732·17-s − 0.0782·18-s + 1.92·19-s + 1.22·20-s − 0.748·21-s − 0.358·22-s − 0.777·23-s − 0.263·24-s + 0.689·25-s − 0.368·26-s − 0.192·27-s − 1.22·28-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
\(L(6)\) |
\(\approx\) |
\(1.783714336\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783714336\) |
\(L(\frac{13}{2})\) |
|
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 | \( 1 + 10.6T + 2.04e3T^{2} \) |
| 5 | \( 1 + 9.08e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 5.76e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 8.14e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.09e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 4.28e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.07e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.39e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 2.13e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.63e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.02e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 7.88e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.39e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 4.74e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.49e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 2.95e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.73e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.37e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.89e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.68e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.55e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 5.66e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.99e10T + 7.15e21T^{2} \) |
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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
−10.88218870283392194423078747427, −9.587778865578214312725895451127, −8.393433361890738216093368170589, −7.996941842568837174281739198217, −6.62283925346055342373959375212, −5.23142313473153012023287445359, −4.25391372402759609890217067569, −3.65280063011739007376845142182, −1.16971596317183510825801741323, −0.876830350430755677496598983982,
0.876830350430755677496598983982, 1.16971596317183510825801741323, 3.65280063011739007376845142182, 4.25391372402759609890217067569, 5.23142313473153012023287445359, 6.62283925346055342373959375212, 7.996941842568837174281739198217, 8.393433361890738216093368170589, 9.587778865578214312725895451127, 10.88218870283392194423078747427